#  $4D$ AdS Einstein-Gauss-Bonnet black hole with Yang-Mills field and its
thermodynamics

Dharm Veer Singh<EMAIL_ADDRESS>Department of Physics, Institute of
Applied Science and Humanities, GLA University, Mathura, 281406 India.  Benoy
Kumar Singh<EMAIL_ADDRESS>Department of Physics, Institute of Applied
Science and Humanities, GLA University, Mathura, 281406 India.  Sudhaker
Upadhyay<EMAIL_ADDRESS>Department of Physics, K. L. S. College,
Nawada, Bihar 805110, India  Department of Physics, Magadh University, Bodh
Gaya, Bihar 824234, India  Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Pune,
Maharashtra 411007, India  School of Physics, Damghan University, P.O. Box
3671641167,
Damghan, Iran

###### Abstract

We derive an exact black hole solution for the Einstein-Gauss-Bonnet gravity
with Yang-Mills field in $4D$ AdS spacetime and investigate its thermodynamic
properties to calculate exact expressions for the black hole mass,
temperature, entropy and heat capacity. The thermodynamic quantities get
modification in the presence of Yang-Mills field, however, entropy remains
unaffected by the Yang-Mills charge. The solution exhibits $P\\--v$
criticality and belongs to the universality class of Van der Waals fluid. We
study the effect of Gauss-Bonnet coupling and Yang-Mills charge on the
critical behaviour and black hole phase transition. We observe that the values
of critical exponents increase with the Yang-Mills charge and decrease with
the Gauss-Bonnet coupling constant.

$4D$ AdS Einstein-Gauss-Bonnet gravity; Black hole solution; Thermodynamics.

##  I Introduction

Einstein theory of general relativity (GR) is one of the most successful
theories in theoretical physics. Despite its appreciable achievements, there
are still some unsolved problems in the universe such as the hierarchy
problem, the cosmological constant problem, and the late time accelerated
expansion of the Universe. This reflects that GR is not the ultimate theory
and needs further generalization. One of the possible generalization for the
standard Einstein-Hilbert action is by adding the higher curvature terms.
Natural candidates for the higher curvature corrections are provided by
Lovelock gravities, which are the unique theories that give rise to generally
covariant field equations lav . Gauss-Bonnet gravity, also called as Einstein-
Gauss-Bonnet (EGB), is the simplest extension of the Einstein-Hilbert action
to include higher curvature Lovelock terms. A Gauss-Bonnet term appears
naturally in the low energy effective action of string theory zwe .

Thermodynamic properties of black holes have been studied for many years hen ;
sud4 ; sud5 ; sud6 ; sud7  and found that black hole spacetime can not only be
assigned standard thermodynamic variables such as temperature or entropy, but
were also shown to possess rich phase structures and admit critical phenomena.
Hawking and Page haw  were the first who studied the thermodynamic properties
of AdS black holes and found the existence of a certain phase transition in
the phase space of the (nonrotating uncharged) Schwarzschild-AdS black hole.
This was the first order phase transition between thermal AdS space and the
Schwarzschild-AdS black hole, with the latter becoming thermodynamically
preferred above a certain critical temperature. The study of the phase
transitions in higher curvature gravity are subject of present research 05 ;
06 ; 07 . For instance, Van der Waals behaviour, (multiple)-rentrant phase
transitions, critical points and isolated critical points for a variety of
asymptotically AdS black holes can be found in Refs. 08 ; 09 ; 10 ; 11 ; 12 ;
13 . Although there various black hole solutions exist, but the black hole
solution for the EGB gravity with Yang-Mills field and their thermal
properties remain unstudied. This provides us an opportunity to bridge this
gap.

Recently, Glavan and Lin gla  introduced the 4-dimensional theory of gravity
with Gauss-Bonnet correction by re-scaling the Gauss-Bonnet coupling
$\alpha\to\alpha/D-4$. Here, it is important to note that taking the limit
$D\rightarrow 4$ after $\alpha\to\alpha/D-4$ either breaks part of
diffeomorphism or leads to extra gravitational degrees of freedom. Clearly,
both of these violate conditions of the Lovelock’s theorem lav . A consistent
theory of $D\rightarrow 4$ EGB gravity with two dynamical degrees of freedom
is proposed by breaking the temporal diffeomorphism invariance Ao1 ; Ao2 . The
generalization to other spherically symmetric black holes has also discussed
in Refs. fran ; Singh:2020nwo ; Hennigar:2020lsl . Other probes include
studies of the regular black holes Singh:2020mty ; Yang:2020jno ;
Kumar:2020uyz ; 32 ; 33 ; 35 ; 36 ; 37 , quasi-normal modes Devi:2020uac ;
Mishra:2020gce , gravitational lensing Jin:2020emq ; Islam:2020xmy ,
thermodynamic properties, $P-v$ criticality and phase transition
Konoplya:2020cbv ; Wei:2020poh ; Singh:2020xju ; Zhang:2020obn ; Singh20 ;
Singh21 ; hendi19 ; hendi20 ; hendi18 ; hendi17 ; hendi16 ; he ; hendi2017 .

In this work, we first obtain a static spherically symmetric black hole
solution for EGB gravity with Yang-Mills field. The solution by Glavan and Lin
can be recovered from this generalized solution. Actually, the resulting
solution interpolates with the Glavan and Lin solution gla  in the Yang-Mills
charge. In order to analyse thermodynamic property, we estimate the mass,
Hawking temperature from area-law and entropy which satisfy the first-law of
thermodynamics. The stability of the system is studied by computing the heat
capacity. Here, we observe that the heat capacity curve is discontinuous at
the critical radius. Further, we notice that there is a flip of sign in the
heat capacity around critical radius. Thus, we can say that $4D$ EGB AdS Yang-
Mills black holes whose horizon radius is smaller than the critical radius are
thermodynamically stable, otherwise it is thermodynamically unstable. Thus,
there is a phase transition occurs at critical horizon radius. The phase
transition occurs from the higher to lower mass black holes. The critical
radius increases along with parameter Gauss-Bonnet coupling constant. The
large value of Yang-Mills charge makes the black hole stable. Moreover, we
compute the critical exponents, a universal property of phase transition,
which exactly match with the mean field theory. Finally, we study the Gibbs
free energy which analyses the phase transition of the black holes analogues
with the Van der Waals phase transition and find that Yang-Mills charge $\nu$
and Gauss-Bonnet coupling constant $\alpha$ affect the Gibbs free energy of
the system.

The paper is organized in seven sections. In section II, we obtain a black
hole solution for the EGB action in the presence of Yang-Mills field. Section
III outlines the first law of thermodynamics and the stability of black hole.
The black hole is considered as the Van der Waals fluid in section IV. Here,
critical behaviour of the model and behaviour of Gibbs free energy are
calculated in this section. The values of critical exponents are estimated in
section V. The discussions and final comments are made in the last section.

##  II Black Hole Solution for Einstein-Gauss-Bonnet gravity in $4D$ AdS
specetime

The Einstein-Gauss-Bonnet (EGB) gravity recently formulated in four dimensions
($4D$) by Glavan and Lin gla  as of higher dimension field equation by
rescaling the coupling constant. The consistent realization of the idea
suggested in Refs. Ao1 ; Ao2  either breaks the part of diffeomorphism or
leads to extra gravitational degrees of freedom. A consistent realization of
$4D$ EGB gravity which is invariant under spatial diffeomorphism and breaks
the time diffeomorphism is proposed by the Aoki, Gorji and Mukhhyama (AGM) Ao1
; Ao2 . In order to get consistent theory of $4D$ Einstein-Gauss-Bonnet
gravity, we should start with Arnowitt-Deser-Misner (ADM) metric Ao1 ; Ao2

$ds^{2}=-n^{2}dt^{2}+\gamma_{ij}(dx^{i}+n^{j}dt)(dx^{j}+n^{j}dt),$ (1)

where $n$, $n_{i}$ and $\gamma_{ij}$ correspond to lapse function, the shift
vector and the spatial metric, respectively.

The action for EGB gravity theory (in natural unit) is given by Ao2

$S=\frac{1}{16\pi}\int dtd^{3}xn\sqrt{\gamma}\left[2{R}-{\cal
M}+{\alpha}R^{2}_{GB}\right].$ (2)

Here $R$ is the Ricci scalar of the “spatial metric”, $\alpha$ is the Gauss-
Bonnet coupling coefficient $\alpha\geq 0$ and Gauss-Bonnet Lagrangian is
defined as

$R^{2}_{GB}=\frac{1}{2}\left[8{R}^{2}-4{R}{\cal M}-{\cal
M}^{2}\right]-\frac{8}{3}\left[8{R}_{ij}{R}^{ij}-4{R}_{ij}{\cal M}^{ij}-{\cal
M}_{ij}{\cal M}^{ij}\right],$ (3)

where ${\cal M}$ is the trace of ${\cal M}_{ij}$ defined as

${\cal M}_{ij}={R}_{ij}+{\cal K}_{\ k}^{k}{\cal K}_{ij}-{\cal K}_{ik}{\cal
K}^{k}_{\ j}\quad\text{with}\quad{\cal
K}_{ij}=\frac{1}{2n}\left[\dot{\gamma}_{ij}-2D_{(i}{n}_{j)}-\gamma_{ij}D^{2}\lambda_{GF}\right].$
(4)

Here, $R$, $R_{ij}$ and ${\cal K}_{ij}$ are the Ricci scalar, Ricci tensor and
extrinsic curvature respectively, the covariant derivative compatible with the
spatial metric is denoted by $D_{i}$ and Lagrange multiplier related to the
gauge-fixing constraint is denoted by $\lambda_{GF}$. One of the properties of
the consistent AGM theory described by the action (2) is that the
$d\rightarrow 3$ limit of the $d+1$-dimensional solution of EGB gravity is the
solution of the AGM theory if it has vanishing Weyl tensor of the spatial
metric in $d+1$ dimensions.

Since, our main objective here is to study a black hole solution with Yang-
Mills field and their thermal properties in $4D$ EGB gravity. Therefore, it is
worth writing the Yang-Mills Lagrangian as

${\cal F}_{YM}=-F_{\mu\nu}^{(a)}F^{(a)\mu\nu},$ (5)

where field-strength tensor is defined in terms of gauge potential ${\cal
A}_{\mu}^{(a)}$ as $F_{\mu\nu}^{(a)}=2\nabla_{[\mu}{\cal
A}_{\nu]}^{(a)}+f_{(b)(c)}^{(a)}{\cal A}_{\mu}^{(b)}{\cal A}_{\nu}^{(c)}$,
with the structure constant $f_{(b)(c)}^{(a)}$ which can be calculated with
the generator of gauge group $SU(2)$.

In order to obtain static spherically symmetric black hole solution of the EGB
gravity with Yang-Mills field (given in Appendix A), we write the following
line element:

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega_{D-2},$ (6)

where $d\Omega_{D-2}$ is the metric of a $(D-2)$-dimensional sphere $S^{D-2}$.
Comparing this metric with the ADM metric (1), one confirms that

$\displaystyle n^{2}=f(r),\qquad
n^{i}=0,\qquad\text{and}\qquad\gamma_{ij}=\mbox{diag}\left(\frac{1}{f(r)},r^{2},r^{2}\sin^{2}\theta,\ldots\right).$
(7)

In order to specify $f(r)$, we appoint the position dependent generators
${\cal T}_{(r)}$, ${\cal T}_{(\theta)}$ and ${\cal T}_{(\phi)}$ of the gauge
group rather the standard one $t_{(1)}$, $t_{(2)}$ and $t_{(3)}$. The
relations between the basis of $SU(2)$ group and the standard basis are given
as

$\displaystyle{\cal T}_{(r)}=\sin\theta\cos\nu\phi
t_{(1)}+\sin\theta\sin\nu\phi t_{(2)}+\cos\theta t_{(3)},$ (8)
$\displaystyle{\cal T}_{(\theta)}=\cos\theta\cos\nu\phi
t_{(1)}+\cos\theta\sin\nu\phi t_{(2)}-\sin\theta t_{(3)},$ (9)
$\displaystyle{\cal T}_{(\phi)}=-\sin\nu\phi t_{(1)}+\cos\nu\phi t_{(2)}.$
(10)

These generators satisfy the following commutation relations:

$\displaystyle\left[{\cal T}_{(r)},{\cal T}_{(\theta)}\right]={\cal
T}_{(\phi)},\qquad\qquad\left[{\cal T}_{(\phi)},{\cal T}_{(r)}\right]={\cal
T}_{(\theta)},\qquad\qquad\left[{\cal T}_{(\theta)},{\cal
T}_{(\phi)}\right]={\cal T}_{(r)}.$ (11)

Meanwhile, we consider the Wu-Yang choice of the gauge potential having the
following non-zero components bal :

$\displaystyle{\cal
A}_{\theta}^{(a)}=\delta_{(\phi)}^{(a)}\quad\text{and}\qquad{\cal
A}_{\phi}^{(a)}=-\nu\sin\theta\delta_{(\theta)}^{(a)}.$ (12)

The Wu-Yang ansatz and the Yang-Mills tensor field lead to the following non-
vanishing component of Yang-Mills field $F_{\theta\phi}^{(r)}=\nu\sin\theta$.
The $(r,r)$ components of the Eq. (54) in the limit $D\to 4$ becomes

$r^{5}-2r^{3}\alpha(f(r)-1)f^{\prime}(r)+r^{4}(f(r)-1)+r^{2}\alpha(f(r)-1)^{2}+\Lambda
r^{2}+\frac{\nu}{r^{2}}=0.$ (13)

For $D\to 4$, we have following solution

$\displaystyle
f_{\pm}(r)=1+\frac{r^{2}}{2\alpha}\left(1\pm\sqrt{1+4\alpha\left(\frac{2M}{r^{3}}-\frac{\nu^{2}}{r^{4}}-\frac{1}{l^{2}}\right)}\,\right),$
(14)

where it is natural to adopt $\Lambda$ in terms scale length $l$ as
$\Lambda=-3/l^{2}$ for AdS solution. This is an exact solution corresponds to
the two branch of solution depending on the choice of signature. Here, it is
worth mentioning that although one can define the local notions like trapped
surfaces and so on, defining the global quantities like event horizon is
ambiguous. The $+ve$ signature is not physical as it reduces to
Reissner–Nordström black hole with negative mass and imaginary charge.
However, $-ve$ signature reduces to the Schwarzschild solution for $\alpha\to
0$, $\nu\to 0$ and $l\to\infty$. Therefore, we limit ourselves to the $-ve$
branch of solution throughout the manuscript. In this solution, $M$ is the
integration constant related to the mass of the black hole, $\alpha$ is the
Gauss-Bonnet coupling and $\nu$ is Yang-Mills charge related to the hair. In
the limit of $\nu=0$, the solution (14) reduces to the $4D$ EGB AdS black hole
solution

$\displaystyle
f(r)=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+4\alpha\left(\frac{2M}{r^{3}}-\frac{1}{l^{2}}\right)}\,\right).$
(15)

The solution by Glavan and Lin gla  can be recovered from this solution in the
limit of $l\to\infty$. Also, the solution (14) reduces to AdS Schwarzschild
black hole when $\nu=0$ and $\alpha\to 0$. This describes AdS Schwarzschild
Yang-Mills black hole when $\alpha\to 0$ with following metric:

$f(r)=1-\frac{2M}{r}+\frac{\nu^{2}}{r^{2}}+\frac{r^{2}}{l^{2}}.$ (16)

As we know that $f(r)=0$ gives the horizons of the black hole, but Eq. (14) is
a transcendental equation and therefore can not be solved analytically. So, we
plot the graph to estimate the horizons as shown in the Fig. 1.

---
Figure 1: The plots of $f(r)$ vs $r$ for different values of Yang-Mills charge $\nu$ for $-ve$ branch of solution (14) with fixed value of mass $(M=1)$ and ${l=20}$ for different Gauss-Bonnet coupling parameter: $\alpha=0.1$ (left) and $\alpha=0.2$ (right).  | $\alpha=0.1$ |  |  |  |  | $\alpha=0.2$ |
---|---|---|---|---|---|---|---
$\nu$ | $r_{-}$ | $r_{+}$ | $\delta$ | $\nu$ | $r_{-}$ | $r_{+}$ | $\delta$
0.75 | 0.419 |  1.567 | 1.146 | 0.70 | 0.443 |  1.543 | 1.100
0.85 | 0.579 |  1.409 | 0.830 | 0.80 | 0.600 |  1.388 | 0.788
0.95$(\nu_{c})$ | 0.995 |  0.995 | 0 | 0.90$(\nu_{c})$ | 0.995 |  0.995 | 0
Table 1: Inner horizon ($r_{-}$), outer horizon ($r_{+}$) and
$\delta=r_{+}-r_{-}$ for different values of Yang-Mills charge $\nu$ with
fixed value of mass $(M=1)$ and AdS length scale ${l=20}$.

In table 1, we identify the different values of the parameters. Here, $r_{+}$
corresponds to the outer horizon while $r_{-}$ represents the inner horizon.
Elementary analysis of the zeros of $f(r)=0$ reveals that it has no zeros if
$\nu>\nu_{c}$, one double zero if $\nu=\nu_{c}$, and two simple zeros if
$\nu<\nu_{c}$, (Fig. 1). From the figure, it is evident that the two horizons
coincide at the critical radius

$r_{c}=\sqrt{\frac{l^{2}\sqrt{1+\frac{12}{l^{2}}(\alpha+\nu^{2})}-l^{2}}{6}}.$
(17)

These cases, therefore, describe $4D$ EGB Yang-Mills black hole with
degenerate horizon and a non-extreme black hole with both outer and inner
Killing horizons, respectively. It is clear that the critical value of $M_{c}$
and $r_{c}$ depend upon the parameters $\alpha$ and $\nu$. Also, the radius of
the outer horizon increases with decrease in the Yang-Mills charge $\nu$ and
the Gauss-Bonnet coupling $\alpha$ as shown in Fig. 1 and Table 1.

From the standard relation $f(r_{+})=0$, the mass parameter of $4D$ EGB AdS
Yang-Mills black hole is calculated as

$M_{+}=\frac{r_{+}}{2}\left(\frac{r_{+}^{2}+\alpha+\nu^{2}}{r_{+}^{2}}+\frac{r_{+}^{2}}{l^{2}}\right).$
(18)

From this expression, we can study various thermodynamical quantities of the
system.

##  III Thermodynamics and stability of black hole

In this section, we compute essential thermodynamic quantities which help in
checking first-law of thermodynamics and the stability of black holes. In
particular we derive Hawking temperature, entropy and heat capacity,
chronologically.

The Hawking temperature associated with the black hole is defined in term of
horizon radius as following:

$T_{+}=\frac{\kappa}{2\pi}=\frac{1}{4\pi}f^{\prime}(r)|_{r=r_{+}},$ (19)

where parameter $\kappa$ refers to the surface gravity.

In our case, the above definition leads to the following value for
temperature:

$T_{+}=\frac{1}{4\pi
r_{+}}\left(\frac{r_{+}^{2}-\alpha-\nu^{2}}{r_{+}^{2}+2\alpha}+\frac{3r_{+}^{4}}{l^{2}(r_{+}^{2}+2\alpha)}\right).$
(20)

For vanishing Yang-Mills charge $\nu=0$, this temperature (20) reduces to the
case of the $4D$ AdS EGB black hole,

$T_{+}=\frac{1}{4\pi
r_{+}}\left(\frac{r_{+}^{2}-\alpha}{r_{+}^{2}+2\alpha}+\frac{3r_{+}^{4}}{l^{2}(r_{+}^{2}+2\alpha)}\right).$
(21)

The relation (20) also reduces to the case of AdS Schwarzschild Yang-Mills
black hole and AdS Schwarzschild black hole for $\alpha=0$ and
$\nu=0,\alpha=0$, respectively.

Now, in order to study the behaviour of temperature, we plot temperature with
respect to event horizon for different values of Yang-Mills charge as shown in
Fig. 2.

---
Figure 2: The plot of temperature $T$ vs $r_{+}$ for different values of Yang-
Mills charge $\nu$ for Gauss-Bonnet coupling constant $\alpha=0.1$ (left) and
$\alpha=0.2$ (right) with fixed value of $l=20$.

From the figure, this is obvious that the Hawking temperature of the $4D$ EGB
AdS Yang-Mills black hole grows to a maximum (say $T_{max}$) and then drops to
minimum temperature. It turns out that the maximum value of the Hawking
temperature decreases with increase of the Yang-Mills charge $\nu$ and Gauss-
Bonnet coupling constant $\alpha$. The Hawking temperature vanishes at the
critical radius shown in Table 2.

|  |  | $\alpha=0.1$ |  |  |  |  |  $\alpha=0.2$ |  |
---|---|---|---|---|---|---|---|---|---|---
$\nu$ |  | 0.75 | 0.85 | 0.95 | 1.05 |  | 0.75 | 0.85 | 0.95 | 1.05
$r_{c}^{T}$ |  |  1.47 | 1.63 |  1.82 | 1.90 |  | 1.67 |  1.77 |  1.92 |  2.52
$T_{+}^{Max}$ |  |  0.0354 | 0.323 |  0.258 | 0.230 |  | 0.0311 |  0.0290 |  0.0270 |  0.256
Table 2: The maximum Hawking temperature ($T_{+}^{max}$) at critical radius
($r_{c}^{T}$) for different values of Yang-Mills charge ($\nu$) with fixed
value ${l=20}$.

Now, we calculate another useful quantity associated with the black hole known
as entropy $S_{+}$. This black hole can be considered as a thermodynamic
system only if the quantities associated with it must obey the first-law of
thermodynamics given below

$\displaystyle dM_{+}=T_{+}dS_{+}+\Phi\,d\nu,$ (22)

where $\Phi$ is the potential of the black hole. It is matter of calculation
only to derive expression of entropy thermo ; Singh:2020xju  from the
expression (22) at constant $\nu$ as

$S_{+}=\pi\left(r_{+}^{2}+4\alpha\log[r_{+}]\right).$ (23)

Finally, we analyse how the background Yang-Mills field affects the
thermodynamic stability of the $4D$ EGB AdS black hole by investigating the
heat capacity $C_{+}$. The stability of the black hole can be estimated from
sign of the heat capacity $C_{+}$ as positive heat capacity reflects the
stable black hole and negative reflects the unstable one.

The heat capacity of the black hole can be estimated from the following
standard relation cai :

$C_{+}=\frac{\partial M_{+}}{\partial T_{+}}=\left(\frac{\partial
M_{+}}{\partial r_{+}}\right)\left(\frac{\partial r_{+}}{\partial
T_{+}}\right).$ (24)

This leads to

$C_{+}=\frac{2\pi(r_{+}^{2}+2\alpha)^{2}(3r_{+}^{4}+l^{2}(r_{+}^{2}-\alpha-\nu^{2}))}{3r_{+}^{4}(r_{+}^{2}+6\alpha)-l^{2}(r_{+}^{4}+2\alpha(\alpha+\nu^{2})-r_{+}^{2}(5\alpha+3\nu^{2}))}.$
(25)

From this expression, it is clearly evident that in the absence of Yang-Mills
charge ($\nu$) the heat capacity (25) reduces to

$C_{+}=\frac{2\pi(r_{+}^{2}+2\alpha)^{2}(3r_{+}^{4}+l^{2}(r_{+}^{2}-\alpha))}{3r_{+}^{4}(r_{+}^{2}+6\alpha)-l^{2}(r_{+}^{4}+2\alpha^{2}-5r_{+}^{2}\alpha)},$
(26)

which further reduces to the heat capacity of the AdS Schwarzschild black hole
in the limit of $\alpha\to 0$. Moreover, Eq. (25) reduces to the heat capacity
of the AdS Schwarzschild Yang-Mills black hole when the Gauss-Bonnet coupling
switched off ($\alpha\to 0$).

---
Figure 3: The plot of temperature $T$ vs $r_{+}$ for different values of Yang-
Mills charge $\nu$ with fixed value of mass $(M=1)$ and ${l=20}$ for Gauss-
Bonnet coupling parameter $\alpha=0.1$ (left) and $\alpha=0.2$ (right).

In order to identify the stability of the heat capacity, we plot expression
(25) for different values of Yang-Mills charge $\nu$ and Gauss-Bonnet coupling
$\alpha$ as shown in Fig. 3. From the plot, we observe that the heat capacity
curve is discontinuous where the temperature is maximum (see Fig. 2 and table
2). The plot clearly shows that the heat capacity diverges at the critical
points $r_{+}^{a}$ and $r_{+}^{b}$ with $r_{+}^{a}<r_{+}^{b}$ (see Fig. 3).
The $4D$ AdS EGB Yang-Mills black hole is stable for the horizon radius
$r_{+}<r_{+}^{a}$ and $r_{+}>r_{+}^{b}$ and unstable for horizon radius
$r_{+}^{a}<r_{+}<r_{+}^{b}$. It is obvious from this figure that $4D$ EGB AdS
Yang-Mills black hole undergoes phase transition twice firstly at $r_{+}^{a}$
from the stable black hole to the unstable black hole and later at $r_{+}^{b}$
from the unstable black hole ($r_{+}^{a}<r_{+}<r_{+}^{b}$) to stable black
hole ($r_{+}>r_{+}^{b}$). Thus, there is a phase transition occurs at
$r_{+}=r_{+}^{a}$ and $r_{+}=r_{+}^{b}$ where black hole changes its phase
from the stable to unstable phases and vice versa. Further, a divergence of
the heat capacity at critical $r_{+}=r_{c}$ signals that this is a second-
order phase transition hp ; davis77 .

The heat capacity is discontinuous at $r_{+}=1.47$, at which the Hawking
temperature has the maximum value $T_{+}=$ for $\alpha=0.1$ and $\nu=0.75$
(Fig. 3). The phase transition occurs from the higher to lower mass black
holes corresponding to positive to negative heat capacity. The critical radius
$r_{c}$ increases along with parameter $\alpha$ (cf. Fig. 3 and Table 2).

##  IV Critical behaviour and phase transition

In the extended phase space the cosmological constant $\Lambda$ is considered
as the thermodynamical pressure in order to have Van der Waals like phase
transition. The relation between the thermodynamical pressure and cosmological
constant is quite standard and given by

$P=-\frac{\Lambda}{8\pi}=\frac{3}{8\pi l^{2}}.$ (27)

Here, units are set such that $G=\hbar=c=1$. The pressure and specific volume
can be calculated as following:

$P=\frac{T_{+}(r_{+}^{2}+2\alpha)}{2r_{+}^{3}}-\frac{r_{+}^{2}-\alpha-\nu^{2}}{8\pi
r_{+}^{4}},\qquad v=2r_{+}.$ (28)

---
Figure 4: The plot of pressure vs specific volume for $T<T_{c}$, $T=T_{c}$ and
$T>T_{c}$ with $T_{c}=0.04038$

In order to study critical behaviour, we plot a $P\\--v$ diagram for the $4D$
EGB AdS Yang-Mills black hole as displayed in Fig. 4. Here, we clearly see
that a small-large black hole phase transition occurs when the temperature of
black hole is less than the critical temperature. To visualize this more
clearly, a transition is directly presented on the right panel of Fig. 4.
Notice that the $P\\--v$ curve contains two stable regions and one unstable
region. These stable regions occur for $v\in[0,\alpha]$ and
$v\in[\beta,\infty]$ and correspond to the small black hole region and the
large black hole region, respectively.

The unstable region $v\in[\alpha,\beta]$ is referred as the spinodal region.
In this unstable region, the small and large black hole phase transition can
coexist. Furthermore, we note that $\partial P(v,T_{0})/\partial v<0$ for the
stable regions, however $\partial P(v,T_{0})/\partial v>0$ for the unstable
region.

Mathematically, the two extreme points $\alpha$ and $\beta$ are determined by
the relation

$\left.\frac{\partial P(v,T_{0})}{\partial
v}\right|_{\nu=\alpha}=\left.\frac{\partial P(v,T_{0})}{\partial
v}\right|_{\nu=\beta}.$ (29)

Moreover, the inflection point at $\nu=\nu_{0}$ is determined by the relation

$\left.\frac{\partial^{2}P(v,T_{0})}{\partial v^{2}}\right|_{T}=0.$ (30)

The critical exponents occur at points of inflections in the $P-v$ diagrams

$\left.\frac{\partial P_{+}}{\partial
r_{+}}\right|_{T}=0,\qquad\qquad\left.\frac{\partial^{2}P_{+}}{\partial
r_{+}^{2}}\right|_{T}=0.$ (31)

This leads to the following values of critical radius, temperature and
pressure, respectively rbm ; guna

$\displaystyle r_{c}=\sqrt{6\alpha+3\nu^{2}+X},$ (32) $\displaystyle
T_{c}=\frac{4\alpha+\nu^{2}+X}{2\pi(\sqrt{6\alpha+3\nu^{2}+X})({12\alpha+3\nu^{2}+X})},$
(33) $\displaystyle
P_{c}=\frac{52\alpha^{2}+\nu^{2}(9\nu^{2}+49\alpha)+X(3\nu^{2}+7\alpha)}{8\pi({6\alpha+3\nu^{2}+X})^{2}({12\alpha+3\nu^{2}+X})},$
(34)

where $X=(48\alpha^{2}+48\alpha\nu^{2}+9\nu^{4})^{1/2}.$

These critical values together with relation $v_{c}=2r_{c}$ give the following
universal ratio:

$\rho_{c}=\frac{P_{c}v_{c}}{T_{c}}=\frac{52\alpha^{2}+\nu^{2}(9\nu^{2}+49\alpha)+X(3\nu^{2}+49\alpha)}{2(6\alpha+3\nu^{2}+X)(4\alpha+\nu^{2}+X)}.$
(35)

In the limit of $\alpha\to 0$ the Eq. (35) resembles to the universal ratio
$\rho_{c}=3/8$ rbm . The Universal ratio of the $4D$ AdS EGB black hole
$\rho_{c}=0.350$ is slightly smaller than the Van der Waals ratio
$\rho_{c}=0.375$ at the small value of $\alpha$ and $\nu$ and the universal
ratio increases with increasing values of $\alpha$ and $\nu$ Hegde:2020xlv .

In order to search for the phase structure of the black hole solutions, we
derive the free energy in the canonical ensemble. The standard definition for
Gibbs free energy reads: $G_{+}=M_{+}-T_{+}S_{+}$ Singh:2020xju . This leads
to

$\displaystyle G_{+}$ $\displaystyle=$ $\displaystyle\frac{3r_{+}^{2}+8\pi
Pr_{+}^{4}+3\alpha+3\nu^{2}}{6r_{+}}-\frac{(8\pi
Pr_{+}^{4}+r_{+}^{2}-\alpha-\nu^{2})(r_{+}^{2}+4\alpha\log
r_{+})}{4r_{+}(r_{+}^{2}+2\alpha)}.$ (36)

In order to analyse the behaviour of Gibbs free energy, we plot the above
expression in terms of temperature. The graph of Gibbs free energy versus
temperature for $\alpha=0.1$ and $\alpha=0.2$ with the fixed value of Yang
Mills charge $\nu$ is given plotted in Fig. 5.

---
Figure 5:  Gibbs free energy vs temperature of $4D$ EGB AdS Yang-Mills black
hole for the fixed value of Yang-Mills charge $\nu$. Left: $\alpha=0.1$.
Right: $\alpha=0.2$.

The Gibbs free energy which analyses the phase transition of the black holes
analogues with the Van der Waals phase transition has the effect of Yang-Mills
charge $\nu$ and Gauss-Bonnet coupling constant $\alpha$ on the phase
structure of the system. First, we fix the Yang-Mills charge $\nu$ and vary
the Gauss-Bonnet coupling constant $\alpha$. The critical pressure and
critical temperature increase with critical radius (see Fig. 5) with Yang-
Mills charge $\nu$. In $G_{+}-T_{+}$ plots the appearance of characteristic
swallow tail shows that the obtained values are critical ones where the phase
transition occurs. In Fig. 5, we can see that swallow tail shape exists when
$P<P_{c}$ for the first order phase transition and $P=P_{c}$ for the second
order phase transition.

##  V Critical Exponents

In this section, we compute the critical exponent, which is a universal
property of phase transition. Usually, near the critical point, a Van der
Waals like phase transition is characterized by the four critical exponents
$\alpha_{0}$, $\beta$, $\gamma$ and $\delta$ describing the behaviour of the
specific heat, order parameter $\eta$, the isothermal compressibility
$\kappa_{T}$ and the critical isotherm, respectively. These are defined as

$\displaystyle C_{v}=T\frac{\partial S}{\partial
T}\propto|\tau|^{-\alpha_{0}},\qquad\qquad\eta=v_{l}-v_{s}\propto|\tau|^{\beta},$
(37) $\displaystyle\kappa_{T}=-\frac{1}{v}\frac{\partial v}{\partial
p}\propto|\tau|^{-\gamma},\quad\qquad|P-P_{c}|_{T_{c}}\propto|v-v_{c}|^{\delta},$
(38)

where $\tau=\frac{T-T_{c}}{T_{c}}$ and $v_{l}$ and $v_{s}$ denote specific
volume of large and small black holes, respectively.

Now, with the following definitions:

$p=\frac{P}{P_{c}},\qquad\qquad
t=\frac{T}{T_{c}}=\tau+1,\qquad\text{and}\qquad\lambda=\frac{v}{v_{c}},$ (39)

the equation of state translate into so-called law of corresponding states and
reads

$\frac{8p}{3}=\frac{t}{\lambda}-\frac{t\alpha_{0}}{6\lambda^{3}}-\frac{}{}\left(\frac{1}{\lambda}+\frac{2\alpha_{0}}{\lambda}+\frac{2\nu^{2}}{36\lambda^{4}}\right).$
(40)

Let us calculate the critical exponents predicted by the corresponding states
(40). We define

$\qquad\qquad\omega=\frac{v-v_{c}}{v_{c}}=\lambda-1.$ (41)

To find the critical exponents we expend equation of state near the critical
points as following:

$p=1+a_{10}\tau-a_{11}\tau\omega-a_{30}\omega^{3}+{\cal
O}(\tau\omega^{2},\omega^{4}),$ (42)

In the absence of Gauss-Bonnet coefficient, the above coefficients resemble to
the Reissner-Nordström black hole. Differentiating the Eq. (42) with respect
to $\omega$ for fixed value of $t$, we get

$dP=P_{c}\left(a_{11}\tau+a_{30}\omega^{2}\right)d\omega.$ (43)

If the black hole undergoes the phase transition from small to large one
keeping temperature and pressure constant while changing the thermodynamic
volume from $\omega_{s}$ to $\omega_{l}$, the equation of state always holds,
i.e.,

$\displaystyle p=1+a_{10}\tau-
a_{11}\tau\omega_{l}-a_{30}\omega_{l}^{3}=1+a_{10}\tau-
a_{11}\tau\omega_{s}-a_{30}\omega_{s}^{3}.$ (44)

Moreover, during the phase transition, the Maxwell’s area law also holds,
i.e.,

$\displaystyle\int_{\omega_{l}}^{\omega_{s}}\omega
dP=\omega\left(a_{11}\tau+a_{30}\omega^{2}\right)dP=0.$ (45)

Solving for the above equations, we obtain a non-trivial solution

$\omega_{l}=-\omega_{s}=\sqrt{-\frac{a_{11}\tau}{a_{30}}}=2\sqrt{-\tau}.$ (46)

Hence we have the order parameter $\eta$ as

$\eta=v_{c}(\omega_{l}-\omega_{s})=\sqrt{-\frac{a_{11}\tau}{a_{30}}}\propto\sqrt{-t}=4v_{c}\sqrt{-\tau}.$
(47)

The relations (37) and (47) confirm that $\beta=1/2$.

The isothermal compressibility $\gamma$ is calculated by

$\kappa_{T}=-\left.\frac{1}{P_{c}(1+\omega)}\frac{\partial\omega}{\partial
P}\right|_{T}\propto\frac{1}{6P_{c}}\frac{1}{\tau}.$ (48)

This implies that $\gamma=1$.

The critical isotherm at $\tau=0$ is estimated from Eq. (42) as $\delta=3$ as
$p-1=-\frac{4}{81}(1-\alpha_{0})\omega^{3}$. Consequently, the four critical
exponents are identified as

$\alpha_{0}=0,\qquad\beta=\frac{1}{2},\qquad\gamma=1\qquad\text{and}\qquad\delta=3.$
(49)

Obviously the entropy is not dependent of temperature and hence the critical
exponent $\alpha_{0}=0$. These critical exponents resembles to the Van der
Walls fluid and these critical exponents satisfy the following thermodynamics
scaling laws

$\displaystyle\alpha_{0}+2\beta+\gamma=2,\qquad\alpha_{0}+\beta(\delta+1)=2,\qquad(1-\alpha_{0})\delta=1+(2-\alpha_{0}),$
(50)
$\displaystyle\gamma(\delta+1)=(2-\alpha_{0})(\delta-1)\qquad\text{and}\qquad\gamma=\beta(\delta-1).$
(51)

We conclude that the thermodynamics exponents associated with the AdS EGB
black hole with Yang-Mills field in the extended phase agree with the mean
field theory prediction rbm ; guna ; lala .

Charged (Abelian) black hole solution with some thermal properties have been
studied so far for the $4D$ AdS EGB model fran ; Singh:2020xju ; Hegde:2020xlv
. Here, we have generalized the black hole solution for $4D$ AdS EGB gravity
model in presence of Yang-Mills (non-Abelian) charge. In fact, we explore the
critical behaviour, phase transition and the critical exponents also along
with thermodynamics of the model. With an appropriate limiting case, one can
recover the solutions of aforementioned references.

##  VI Discussions and Conclusions

We have found an exact solution for $4D$ EGB black hole in the presence of
Yang-Mills field which is a generalization of the solution obtained by Glavan
and Lin gla . Also, one recovers the Schwarzschild AdS solution for
$\alpha\rightarrow 0$, $\nu\rightarrow 0$ and $l\rightarrow\infty$.
Furthermore, we have investigated the thermodynamics behaviour of the
solution. Here, we have observed that the thermodynamics quantities get
modification due to the presence of Yang-Mills charge $\nu$. However, the
entropy of the black hole remains unaffected by the Yang-Mills charge and
turns out to be the Bekenstein-Hawking area-law for the novel $4D$ EGB black
hole. In order to discuss the stability of the black hole we derive the heat
capacity. The heat capacity $C_{+}$ diverges at a critical radius $r=r_{c}$,
where the temperature has a maximum and $C_{+}>0$ for $r_{+}<r_{c}$ allowing
the black hole to become thermodynamically stable.

We also examined the $P\\--v$ criticality and phase structure of the $4D$ AdS
EGB black hole solution with Yang-Mills charge. We have found that a small-
large black hole phase transition occurs for the case where temperature is
less than the critical temperature, i.e., $T<T_{c}$. The critical exponents
are also calculated which increase with the Yang-Mills charge $\nu$ and
decrease with the GB coupling constant $\alpha$. It will be interesting to
study the effect of thermal and geometric fluctuation on the thermodynamics of
this black hole solution. This is the subject of future investigation.

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##  Appendix A The equations of motion for EGB gravity with Yang-Mills field

The action of the $D$-dimensional EGB gravity in the presence of Yang-Mills
field is given by

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{D}x\sqrt{-g}\left[{\cal R}-2\Lambda+\frac{\alpha}{D-4}{\cal L_{GB}}-{\cal
F}_{YM}\right],$ (52)

where $\mathcal{L_{GB}}$ is Lagrangian density for EGB gravity and has
following expression:

$\mathcal{L_{GB}}={\cal R}_{\mu\nu\gamma\delta}{\cal
R}^{\mu\nu\gamma\delta}-4{\cal R}_{\mu\nu}{\cal R}^{\mu\nu}+{\cal R}^{2}.$

The equation of motion for metric tensor $g_{\mu\nu}$ and the electromagnetic
potential ${\cal A}_{\mu}^{(a)}$ are given, respectively, by

$\displaystyle G_{\mu\nu}+\Lambda g_{\mu\nu}+H_{\mu\nu}=T_{\mu\nu}^{YM},$ (53)
$\displaystyle D_{\mu}F^{(a)\mu\nu}=0,$ (54)

where

$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle{\cal
R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{\cal R},$ (55) $\displaystyle H_{\mu\nu}$
$\displaystyle=$ $\displaystyle-\frac{\alpha}{2}\left[8{\cal
R}^{\rho\sigma}{\cal R}_{\mu\rho\nu\sigma}-4{\cal
R}_{\mu}^{\rho\sigma\lambda}{\cal R}_{\nu\rho\sigma\lambda}-4{\cal R}{\cal
R}_{\mu\nu}+8{\cal R}_{\mu\lambda}{\cal R}_{\nu}^{\lambda}\right.$
$\displaystyle+$ $\displaystyle\,\,\left.g_{\mu\nu}\left({\cal
R}_{\rho\sigma\gamma\delta}{\cal R}^{\rho\sigma\gamma\delta}-4{\cal
R}_{\rho\sigma}{\cal R}^{\rho\sigma}+{\cal R}^{2}\right)\right],$
$\displaystyle T_{\mu\nu}^{YM}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}g_{\mu\nu}F_{\rho\sigma}^{(a)}F^{(a)\rho\sigma}+2F_{\mu\sigma}^{(a)}F^{(a)\sigma}_{\
\nu}.$ (56)

Here ${\cal R}$, ${\cal R}_{\mu\nu}$ and ${\cal R}_{\mu\nu\gamma\delta}$ the
$D=d+1$ dimensional Ricci scalar, Ricci tensor and Riemann tensor,
respectively.


# An approximation problem in the space of bounded operators

Arpita Mal  Kolkata 700032
India<EMAIL_ADDRESS>
###### Abstract.

For Banach spaces $X,Y,$ we consider a distance problem in the space of
bounded linear operators $\mathcal{L}(X,Y).$ Motivated by a recent paper [19],
we obtain sufficient conditions so that for a compact operator
$T\in\mathcal{L}(X,Y)$ and a closed subspace $Z\subset Y,$ the following
equation holds, which relates global approximation with local approximation:

$d(T,\mathcal{L}(X,Z))=\sup\\{d(Tx,Z):x\in X,\|x\|=1\\}.$

In some cases, we show that the supremum is attained at an extreme point of
the corresponding unit ball. Furthermore, we obtain some situations when the
following equivalence holds:

$T\perp_{B}\mathcal{L}(X,Z)\Leftrightarrow
T^{**}x_{0}^{**}\perp_{B}Z^{\perp\perp}\Leftrightarrow
T^{**}\perp_{B}\mathcal{L}(X^{**},Z^{\perp\perp}),$

for some $x_{0}^{**}\in X^{**}$ satisfying
$\|T^{**}x_{0}^{**}\|=\|T^{**}\|\|x_{0}^{**}\|,$ where $Z^{\perp}$ is the
annihilator of $Z.$ One such situation is when $Z$ is an $L^{1}-$predual space
and an $M-$ideal in $Y$ and $T$ is a multi-smooth operator of finite order.
Another such situation is when $X$ is an abstract $L_{1}-$space and $T$ is a
multi-smooth operator of finite order. Finally, as a consequence of the
results, we obtain a sufficient condition for proximinality of a subspace $Z$
in $Y.$

###### Key words and phrases:

Best approximation; distance formulae; Birkhoff-James orthogonality; linear
operators; $L^{1}-$predual

###### 2010 Mathematics Subject Classification:

Primary 46B28, Secondary 41A50, 46B25, 47L05

The author would like to thank Prof. Kallol Paul for his valuable comments on
this paper.

##  1\. Introduction

In this paper, our aim is to study distance problem and Birkhoff-James
orthogonality in the space of bounded linear operators. We approximate the
distance of a bounded linear operator from a subspace of operators with the
distance of its image from a subspace. To state the problem, we first
introduce necessary notations and terminologies.

In this paper, $X,Y$ denote real Banach spaces and $Z$ denotes a closed
subspace of $Y.$ Let $S_{X}$ and $B_{X}$ denote the unit sphere and the unit
ball of $X,$ i.e., $S_{X}=\\{x\in X:\|x\|=1\\}$ and $B_{X}=\\{x\in X:\|x\|\leq
1\\}.$ Suppose $E_{X}$ denotes the set of all extreme points of $B_{X}$ and
$X^{*}$ denotes the dual space of $X.$ We always assume that $X$ is
canonically embedded in its bidual $X^{**}.$ The symbol
$\mathcal{L}(X,Y)~{}(\mathcal{K}(X,Y))$ denotes the space of all bounded
(compact) linear operators from $X$ to $Y.$ For $T\in\mathcal{L}(X,Y),$
$M_{T}=\\{x\in S_{X}:\|Tx\|=\|T\|\\},$ the collection of all unit vectors of
$X,$ at which $T$ attains its norm. For a non-zero element $x\in X,$ suppose
$J(x)=\\{x^{*}\in S_{X^{*}}:x^{*}(x)=\|x\|\\}.$ Recall that $J(x)$ is a non-
empty, weak*compact, convex subset of $S_{X^{*}}.$ The symbol $E_{J(T)}$
denotes the set of all extreme points of $J(T).$ A non-zero element $x\in X$
is said to be a multi-smooth point of finite order (or $k-$smooth) [8, 11] if
$J(x)$ contains finitely many (exactly $k$) linearly independent functionals.
In other words, $x$ is a multi-smooth point of finite order if $Span(J(x))$ is
finite-dimensional. We say that $x$ is smooth if $J(x)$ is singleton and $X$
is smooth if $x$ is smooth for all non-zero $x\in X.$ Similarly, a non-zero
operator $T\in\mathcal{L}(X,Y)$ is a multi-smooth operator of finite order if
$Span(J(T))$ is finite-dimensional, where $J(T)=\\{f\in
S_{\mathcal{L}(X,Y)^{*}}:f(T)=\|T\|\\}.$ In particular, $T$ is said to be a
smooth operator if $J(T)$ is singleton. For the study of smooth and multi-
smooth ($k-$smooth) operators see [4, 5, 11, 12, 13, 16, 17, 24].
A classical problem in approximation theory is the distance problem. For $x\in
X$ and a subspace $W\subset X,$ $d(x,W)=\inf_{w\in W}\|x-w\|$ is the distance
of $x$ from $W.$ An element $w_{0}\in W$ is said to be a best approximation to
$x$ out of $W$ if $\|x-w_{0}\|=d(x,W).$ We denote the collection of all best
approximation(s) to $x$ out of $W$ by $\mathscr{L}_{W}(x),$ i.e.,
$\mathscr{L}_{W}(x)=\\{w_{0}\in W:\|x-w_{0}\|=d(x,W)\\}.$ A subspace $W\subset
X$ is said to be a proximinal subspace of $X$ if
$\mathscr{L}_{W}(x)\neq\emptyset$ for all $x\in X.$ Similarly, in operator
space, for $T\in\mathcal{L}(X,Y)$ and a subspace
$\mathscr{V}\subset\mathcal{L}(X,Y),$
$d(T,\mathscr{V})=\inf_{S\in\mathscr{V}}\|T-S\|$ and
$\mathscr{L}_{\mathscr{V}}(T)=\\{S_{0}\in\mathscr{V}:\|T-S_{0}\|=d(T,\mathscr{V})\\}.$
For some recent study on best approximation and distance formula in operator
spaces and $C^{*}$\- algebra follow [3, 14, 21, 22]. Note that, the notions of
best approximation and Birkhoff-James orthogonality are closely related. For
$x,y\in X,$ we say that $x$ is Birkhoff-James orthogonal [2, 7] to $y$ if
$\|x+\lambda y\|\geq\|x\|$ for all scalars $\lambda$ and we denote it by
$x\perp_{B}y.$ For a subspace $W\subset X,$ we say that $x\perp_{B}W$ if
$x\perp_{B}w$ for all $w\in W.$ It is now straightforward to check that
$w_{0}\in\mathscr{L}_{W}(x)$ if and only if $x-w_{0}\perp_{B}W.$ Suppose that
$K$ is a compact Hausdorff space. Let $C(K,Y)$ be the space of all continuous
functions defined from $K$ to $Y$ equipped with the supremum norm. From a
classical result [10, Th. 2.4] by Light and Cheney, we know that if $f\in
C(K,Y),$ then

$d(f,C(K,Z))=\sup_{k\in K}d(f(k),Z)=d(f(k_{0}),Z)$

for some $k_{0}\in K.$ The above distance formula provides a relation between
global approximation and local approximation. Motivated by the above
approximation result, recently in [19], Rao studied analogous problem in the
space of bounded linear operators. More precisely, he raised the question that
for $T\in\mathcal{K}(X,Y),$ when the following minimax formula holds:

(1) $d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z).$

It is also interesting to ask when the supremum in (1) is attained. Whenever
the the supremum is attained at some $x_{0}\in S_{X},$ we can further ask
whether $x_{0}$ can be chosen from $E_{X}.$ To answer these questions, we need
to introduce a few notions. A Banach space $X$ is said to be an
$L^{1}-$predual space [9] if $X^{*}$ is isometrically isomorphic to
$L^{1}(\mu)$ for a positive measure $\mu.$ A subspace $Z$ of $Y$ is said to be
an $M-$ideal [6] in $Y,$ if there is a linear projection $P:Y^{*}\to Y^{*}$
such that $\|y^{*}\|=\|Py^{*}\|+\|y^{*}-Py^{*}\|$ for all $y^{*}\in Y^{*}$ and
$\ker(P)=Z^{\perp},$ where $Z^{\perp}$ is the annihilator of $Z,$ i.e.,
$Z^{\perp}=\\{y^{*}\in Y^{*}:y^{*}(z)=0~{}\forall~{}z\in Z\\}.$
Note that, for any $x\in S_{X}$ and $S\in\mathcal{L}(X,Z),$ $\|Tx-Sx\|\geq
d(Tx,Z).$ Thus,

$\|T-S\|=\sup_{x\in S_{X}}\|Tx-Sx\|\geq\sup_{x\in S_{X}}d(Tx,Z).$

Now, taking infimum over $S\in\mathcal{L}(X,Z),$ we get

(2) $d(T,\mathcal{L}(X,Z))\geq\sup_{x\in S_{X}}d(Tx,Z).$

In [19, Th. 1] Rao proved the existence of the minimax formula (1) with the
attainment of the supremum at an extreme point of $B_{X}$ assuming that $X$ is
a reflexive, separable Banach space, $Z$ is an $L^{1}-$predual space and also
an $M-$ideal in $Y.$ On the other hand, assuming that $X$ is a separable
Banach space, $Z$ is an $L^{1}-$predual space and an $M-$ideal in $Y,$ Rao in
[19, Th. 6] proved that under a local condition (a suitable smoothness
condition) on $T$ the minimax formula (1) holds. Both the proofs of these
theorems are based on a lifting theorem from [6, Th. II.2.1].
Many researchers are devoted to the study of Birkhoff-James orthogonality in
the space of operators (see [1, 15, 22, 23] and the references therein). Note
that, if there exists a vector $x_{0}\in M_{T}$ such that $Tx_{0}\perp_{B}Z,$
then for all $S\in\mathcal{L}(X,Z),$
$\|T-S\|\geq\|Tx_{0}-Sx_{0}\|\geq\|Tx_{0}\|=\|T\|.$ Therefore, in this case,
$T\perp_{B}\mathcal{L}(X,Z).$ On the other hand, observe that if the supremum
in (1) is attained at some $x_{0}\in S_{X}$ and $T\perp_{B}\mathcal{L}(X,Z),$
then from $\|T\|=d(T,\mathcal{L}(X,Z))=d(Tx_{0},Z)\leq\|Tx_{0}\|=\|T\|,$ it
follows that $d(Tx_{0},Z)=\|Tx_{0}\|,$ i.e., $Tx_{0}\perp_{B}Z.$ In [19], Rao
used this approach to prove the implication
$T\perp_{B}\mathcal{L}(X,Z)\Rightarrow Tx_{0}\perp_{B}Z$ for some $x_{0}\in
M_{T},$ whenever the supremum in (1) is attained.
In this paper, we prove the existence of formula (1) whenever $T$ is a multi-
smooth operator of finite order and $T\perp_{B}\mathcal{L}(X,Z).$ In this
case, we do not assume any restriction on the space $X.$ We only assume that
$Z$ is an $L^{1}-$predual space and an $M-$ideal in $Y.$ Moreover, we prove
the following equivalence

(3) $T\perp_{B}\mathcal{L}(X,Z)\Leftrightarrow
T^{**}x_{0}^{**}\perp_{B}Z^{\perp\perp}\Leftrightarrow
T^{**}\perp_{B}\mathcal{L}(X^{**},Z^{\perp\perp})$

for some $x_{0}^{**}\in M_{T^{**}}.$ Furthermore, if $X^{*}$ is assumed to be
smooth, then $x_{0}^{**}$ can be chosen from $M_{T^{**}}\cap E_{X^{**}}$ (see
Theorem 2.2 and Remark 2.8). In addition, if $X$ is assumed to be reflexive,
then we show that the supremum in (1) is attained at some $x_{0}\in M_{T}.$ On
the other hand, we show that (1) holds for an arbitrary closed subspace $Z$ of
$Y,$ if we assume that $X^{*}$ is an $L^{1}-$predual space (more generally, if
$X^{**}$ has $L^{1}-$property according to Definition 2.5), $T$ is a multi-
smooth operator of finite order and $T\perp_{B}\mathcal{L}(X,Z).$ Moreover, we
show that in this case, (3) holds for some $x_{0}^{**}\in M_{T^{**}}\cap
E_{X^{**}}$ (see Theorem 2.6, Remark 2.8 and Remark 2.7). Finally, we provide
another situation when formula (1) holds and the supremum is attained at some
$x_{0}\in M_{T}\cap E_{X}.$ As a consequence of the results, we prove that if
$\mathcal{L}(\ell_{1}^{n},Z)$ is a proximinal subspace of
$\mathcal{L}(\ell_{1}^{n},Y),$ then $Z$ is a proximinal subspace of $Y,$
provided $Z$ is an arbitrary closed subspace of $Y$ and each non-zero element
of $Y$ is a multi-smooth point of finite order. We would like to mention that
the approach used in this paper to prove the non-trivial part is completely
different from [19]. In particular, we do not use the lifting theorem from [6,
Th. II.2.1].

##  2\. Main results

We begin this section with an easy proposition.

######  Proposition 2.1.

Let $X,Y$ be Banach spaces and $Z$ be a closed subspace of $Y.$ Let
$T\in\mathcal{L}(X,Y).$ Suppose there exists
$S\in\mathscr{L}_{\mathcal{L}(X,Z)}(T)$ such that $T-S$ is smooth and
$M_{T-S}\neq\emptyset.$ Then there exist $x_{0}\in E_{X},~{}y_{0}^{*}\in
E_{Y^{*}}\cap Z^{\perp}$ such that the following hold.

$\displaystyle d(T,\mathcal{L}(X,Z))$ $\displaystyle=$
$\displaystyle\sup_{x\in S_{X}}d(Tx,Z)=d(Tx_{0},Z)$ $\displaystyle=$
$\displaystyle\sup\\{y^{*}(Tx):x\in E_{X},y^{*}\in E_{Y^{*}}\cap
Z^{\perp}\\}=y_{0}^{*}(Tx_{0}).$

###### Proof.

From $S\in\mathscr{L}_{\mathcal{L}(X,Z)}(T),$ it follows that
$T_{0}\perp_{B}\mathcal{L}(X,Z),$ where $T_{0}=T-S.$ Since $T_{0}$ is smooth
and $M_{T_{0}}\neq\emptyset,$ from [24, Th. 3.3], we get $M_{T_{0}}=\\{\pm
x_{0}\\}$ for some $x_{0}\in S_{X}.$ Observe that $x_{0}\in E_{X}.$ For
otherwise, there exist $x_{1},x_{2}\in B_{X}$ such that $x_{1}\neq x_{2}$ and
$x_{0}=\frac{1}{2}x_{1}+\frac{1}{2}x_{2}.$ Now, from

$\|T_{0}\|=\|T_{0}x_{0}\|\leq\frac{1}{2}\|T_{0}x_{1}\|+\frac{1}{2}\|T_{0}x_{2}\|\leq\frac{1}{2}\|T_{0}\|+\frac{1}{2}\|T_{0}\|=\|T_{0}\|,$

we get that $\|T_{0}x_{1}\|=\|T_{0}x_{2}\|=\|T_{0}\|,$ i.e., $x_{1},x_{2}\in
M_{T_{0}}.$ Thus, $x_{1}=-x_{2}$ and so $x_{0}=0,$ a contradiction. Choose
$x^{*}\in J(x_{0}).$ Let $z\in Z.$ Consider $A\in\mathcal{L}(X,Z)$ defined as
$Ax=x^{*}(x)z$ for all $x\in X.$ Then $T_{0}\perp_{B}A.$ By [24, Th. 3.3], we
get $T_{0}x_{0}\perp_{B}Ax_{0}\Rightarrow T_{0}x_{0}\perp_{B}z.$ Since $z\in
Z$ is chosen arbitrarily, we have $T_{0}x_{0}\perp_{B}Z.$ Thus,

$d(Tx_{0},Z)=d(T_{0}x_{0},Z)=\|T_{0}x_{0}\|=\|T_{0}\|=d(T_{0},\mathcal{L}(X,Z))=d(T,\mathcal{L}(X,Z)).$

On the other hand,

(4) $\sup_{x\in S_{X}}d(Tx,Z)\geq
d(Tx_{0},Z)=d(T,\mathcal{L}(X,Z))\geq\sup_{x\in S_{X}}d(Tx,Z),$

where the last inequality follows from (2). This completes the proof of the
first part.
Now, we prove the second part. Since $T_{0}$ is smooth, again from [24, Th.
3.3], we get, $T_{0}x_{0}$ is smooth, i.e., $J(T_{0}x_{0})=\\{y_{0}^{*}\\}$
for some $y_{0}^{*}\in S_{Y^{*}}.$ Since $J(T_{0}x_{0})$ is convex,
$y_{0}^{*}$ is an extreme point of $J(T_{0}x_{0}).$ Observe that
$J(T_{0}x_{0})$ is an extremal subset of $B_{Y^{*}}.$ Therefore, $y_{0}^{*}\in
E_{Y^{*}}.$ Now, from [7, Th. 2.1] and $T_{0}x_{0}\perp_{B}Z,$ it follows that
$y_{0}^{*}\in Z^{\perp}.$ Thus,

(5)
$y_{0}^{*}(Tx_{0})=y_{0}^{*}(Tx_{0}-Sx_{0})=y_{0}^{*}(T_{0}x_{0})=\|T_{0}x_{0}\|=\|T_{0}\|=d(T,\mathcal{L}(X,Z)).$

On the other hand, observe that for each $x\in E_{X},$ $y^{*}\in E_{Y^{*}}\cap
Z^{\perp}$ and $A\in\mathcal{L}(X,Z),$ we have

$y^{*}(Tx)=y^{*}(Tx-Ax)\leq\|Tx-Ax\|\leq\|T-A\|.$

Thus,

$\sup\\{y^{*}(Tx):x\in E_{X},y^{*}\in E_{Y^{*}}\cap
Z^{\perp}\\}\leq\inf_{A\in\mathcal{L}(X,Z)}\|T-A\|=d(T,\mathcal{L}(X,Z)).$

The above inequality together with (5) completes the proof of the second part.
∎

To prove the next theorem, we use the extremal structure of the unit ball of
$\mathcal{K}(X,Y)^{*}.$ From [20, Th. 1.3], we note that

(6) $E_{\mathcal{K}(X,Y)^{*}}=\\{x^{**}\otimes y^{*}:x^{**}\in
E_{X^{**}},y^{*}\in E_{Y^{*}}\\},$

where $x^{**}\otimes y^{*}(S)=x^{**}(S^{*}y^{*})$ for $S\in\mathcal{L}(X,Y).$
Now, we are ready to prove our desired theorem.

######  Theorem 2.2.

Let $X,Y$ be Banach spaces. Suppose $Z$ is a subspace of $Y$ such that $Z$ is
an $L^{1}-$predual space and an $M-$ideal in $Y.$ Let $T\in\mathcal{K}(X,Y)$
be a multi-smooth operator of finite order. Suppose that
$T\perp_{B}\mathcal{L}(X,Z).$ Then the following hold.
(i) $d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z).$
(ii) $T^{**}x_{0}^{**}\perp_{B}Z^{\perp\perp}$ for some $x_{0}^{**}\in
M_{T^{**}}.$
(iii) $T^{**}\perp_{B}\mathcal{L}(X^{**},Z^{\perp\perp}).$
(iv) There exists $x_{0}^{**}\in M_{T^{**}}$ such that

$\displaystyle d(T^{**},\mathcal{L}(X^{**},Z^{\perp\perp}))$ $\displaystyle=$
$\displaystyle\sup_{x^{**}\in S_{X^{**}}}d(T^{**}x^{**},Z^{\perp\perp})$
$\displaystyle=$ $\displaystyle
d(T^{**}x_{0}^{**},Z^{\perp\perp})=d(T,\mathcal{L}(X,Z)).$

Additionally, if we assume that $X^{*}$ is smooth, then in (ii) and (iv), we
may choose $x_{0}^{**}$ from $M_{T^{**}}\cap E_{X^{**}}.$

###### Proof.

(i) From $T\perp_{B}\mathcal{L}(X,Z)$ and [7, Th. 2.1], it follows that there
exists $f\in J(T)$ such that $f(A)=0$ for all $A\in\mathcal{L}(X,Z).$ Since
$T$ is a multi-smooth operator of finite order, $Span(E_{J(T)})$ is finite-
dimensional. Now, $J(T)$ being a non-empty, weak*compact, convex set, by the
Krein-Milman theorem, we get

$J(T)=\overline{conv}^{w^{*}}(E_{J(T)})\subseteq\overline{Span}^{w^{*}}(E_{J(T)})=\overline{Span}(E_{J(T)})=Span(E_{J(T)}).$

Using [25, Lem. 1.1, pp. 166], we get extreme points
$f_{1},f_{2},\ldots,f_{h}$ of the unit ball of $Span(E_{J(T)})$ and scalars
$\lambda_{1},\lambda_{2},\ldots,\lambda_{h}>0$ such that
$\sum_{i=1}^{h}\lambda_{i}=1$ and $f=\sum_{i=1}^{h}\lambda_{i}f_{i}.$ Now, it
is easy to check that $f_{i}\in E_{J(T)}$ for all $1\leq i\leq h.$ Since
$J(T)$ is an extremal subset of $B_{\mathcal{K}(X,Y)^{*}},$ each $f_{i}$ is an
extreme point of $B_{\mathcal{K}(X,Y)^{*}}.$ Therefore, there exist
$x_{i}^{**}\in E_{X^{**}},y_{i}^{*}\in E_{Y^{*}}$ such that
$f_{i}=x_{i}^{**}\otimes y_{i}^{*}$ for each $1\leq i\leq h.$ Now, $f_{i}\in
J(T)$ implies that

$\|T\|=f_{i}(T)=x_{i}^{**}\otimes
y_{i}^{*}(T)=x_{i}^{**}(T^{*}y_{i}^{*})\leq\|T^{*}y_{i}^{*}\|\leq\|T^{*}\|=\|T\|,$

which yields that $y_{i}^{*}\in M_{T^{*}}$ and
$x_{i}^{**}(T^{*}y_{i}^{*})=\|T^{*}y_{i}^{*}\|=\|T\|.$ Since $Z$ is an
$M-$ideal in $Y,$ by [6, Rem. 1.13, pp. 11] we have
$Y^{*}=Z^{*}\oplus_{1}Z^{\perp}$ and by [6, Lem. 1.5, pp. 3]
$E_{Y^{*}}=E_{Z^{*}}\cup E_{Z^{\perp}}.$ Thus, $y_{i}^{*}\in E_{Z^{*}}\cup
E_{Z^{\perp}}$ for all $1\leq i\leq h.$ We claim that for some $i,$
$y_{i}^{*}\in E_{Z^{\perp}}.$ If possible, suppose that $y_{i}^{*}\in
E_{Z^{*}}$ for all $i.$ Since $Z$ is an $L^{1}-$predual space, either
$\\{y_{i}^{*}:1\leq i\leq h\\}$ is linearly independent or $y_{i}^{*}=\pm
y_{j}^{*}$ for some $i\neq j.$ In the next two paragraphs, we show that after
suitable modification, we can write
$f=\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}\otimes y_{i}^{*},$ where $y_{i}^{*}\in
E_{Y^{*}}$ and $\\{y_{i}^{*}:1\leq i\leq h\\}$ is linearly independent.

Now, suppose that $X^{*}$ is smooth. Observe that if $y_{i}^{*}=y_{j}^{*}$ for
some $i\neq j,$ then
$x_{i}^{**}(T^{*}y_{i}^{*})=x_{j}^{**}(T^{*}y_{i}^{*})=\|T^{*}y_{i}^{*}\|,$
i.e., $x_{i}^{**},x_{j}^{**}\in J(T^{*}y_{i}^{*}).$ The smoothness of
$T^{*}y_{i}^{*}$ yields that $x_{i}^{**}=x_{j}^{**}.$ In this case,
$\lambda_{i}x_{i}^{**}\otimes y_{i}^{*}+\lambda_{j}x_{j}^{**}\otimes
y_{j}^{*}=(\lambda_{i}+\lambda_{j})x_{i}^{**}\otimes y_{i}^{*}.$ Therefore, in
case $X^{*}$ is smooth, if necessary changing the scalars suitably, we may
write $f=\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}\otimes y_{i}^{*},$ where
$x_{i}^{**}\in E_{X^{**}},y_{i}^{*}\in E_{Y^{*}}$ and $\\{y_{i}^{*}:1\leq
i\leq h\\}$ is linearly independent.

Now, suppose that $X^{*}$ is not smooth. Observe that, if
$y_{1}^{*}=y_{2}^{*}$ holds, then considering
$x^{**}=\lambda_{1}x_{1}^{**}+\lambda_{2}x_{2}^{**},$ we get
$x^{**}(T^{*}y_{1}^{*})=(\lambda_{1}+\lambda_{2})\|T^{*}y_{1}^{*}\|$ and
$\|x^{**}\|=\lambda_{1}+\lambda_{2}.$ In that case,
$f=(\lambda_{1}+\lambda_{2})\frac{x^{**}}{\|x^{**}\|}\otimes
y_{1}^{*}+\lambda_{3}x_{3}^{**}\otimes
y_{3}^{*}+\ldots+\lambda_{h}x_{h}^{**}\otimes y_{h}^{*}$ and $x^{**}$ may not
belong to $E_{X^{**}}.$ Therefore, in case $X^{*}$ is not smooth, if necessary
after suitable change, we may write
$f=\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}\otimes y_{i}^{*},$ where $x_{i}^{**}\in
S_{X^{**}},y_{i}^{*}\in E_{Y^{*}}$ and $\\{y_{i}^{*}:1\leq i\leq h\\}$ is
linearly independent.

Now, choose $x^{*}\in X^{*}$ such that $x_{1}^{**}(x^{*})\neq 0$ and
$z_{0}\in\cap_{i=2}^{h}\ker(y_{i}^{*})\setminus\ker(y_{1}^{*}).$ Define
$A\in\mathcal{L}(X,Z)$ by$A(x)=x^{*}(x)z_{0}$ for all $x\in X.$ Therefore,
$A^{*}:Z^{*}\to X^{*}$ is defined as $A^{*}z^{*}=z^{*}(z_{0})x^{*}$ for all
$z^{*}\in Z^{*}.$ Now, from $f(A)=0,$ it follows that

$\displaystyle\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}\otimes y_{i}^{*}(A)=0$
$\displaystyle\Rightarrow$
$\displaystyle\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}(A^{*}y_{i}^{*})=0$
$\displaystyle\Rightarrow$
$\displaystyle\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}(y_{i}^{*}(z_{0})x^{*})=0$
$\displaystyle\Rightarrow$
$\displaystyle\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}(x^{*})y_{i}^{*}(z_{0})=0$
$\displaystyle\Rightarrow$
$\displaystyle\lambda_{1}x_{1}^{**}(x^{*})y_{1}^{*}(z_{0})=0,$

which is a contradiction. This proves our claim. Thus, we get
$i\in\\{1,2,\ldots,h\\}$ such that

(7) $y_{i}^{*}\in E_{Z^{\perp}}\cap
M_{T^{*}}~{}\text{and~{}}x_{i}^{**}(T^{*}y_{i}^{*})=\|T\|.$

Now, for each $x\in X$ and for each $z\in Z,$

$\|Tx-z\|\geq|y_{i}^{*}(Tx-z)|=|y_{i}^{*}(Tx)|=|T^{*}y_{i}^{*}(x)|.$

Thus, taking infimum over $z\in Z,$ we get for each $x\in X,$

$\|Tx\|\geq d(Tx,Z)\geq|T^{*}y_{i}^{*}(x)|.$

In this inequality, taking supremum over $x\in S_{X},$ we have

$\|T\|\geq\sup_{x\in S_{X}}d(Tx,Z)\geq\sup_{x\in
S_{X}}|T^{*}y_{i}^{*}(x)|=\|T^{*}y_{i}^{*}\|=\|T^{*}\|=\|T\|.$

Therefore, $\|T\|=\sup_{x\in S_{X}}d(Tx,Z).$ On the other hand, from
$T\perp_{B}\mathcal{L}(X,Z),$ it clearly follows that
$d(T,\mathcal{L}(X,Z))=\|T\|.$ This proves (i).

(ii) Let $u\in Z^{\perp\perp}.$ Then using (7), we get $u(y_{i}^{*})=0.$ Thus,
for all $u\in Z^{\perp\perp},$

$\displaystyle\|T^{**}x_{i}^{**}-u\|\geq|(T^{**}x_{i}^{**}-u)y_{i}^{*}|=|T^{**}x_{i}^{**}(y_{i}^{*})|=|x_{i}^{**}(T^{*}y_{i}^{*})|$
$\displaystyle\Rightarrow$ $\displaystyle
d(T^{**}x_{i}^{**},Z^{\perp\perp})\geq|x_{i}^{**}(T^{*}y_{i}^{*})|=\|T\|~{}(\text{taking~{}
infimum~{} over~{}}u\in Z^{\perp\perp})$ $\displaystyle\Rightarrow$
$\displaystyle\|T^{**}\|\geq\|T^{**}x_{i}^{**}\|\geq
d(T^{**}x_{i}^{**},Z^{\perp\perp})\geq\|T\|=\|T^{**}\|$
$\displaystyle\Rightarrow$
$\displaystyle\|T^{**}\|=\|T^{**}x_{i}^{**}\|=d(T^{**}x_{i}^{**},Z^{\perp\perp})$
$\displaystyle\Rightarrow$ $\displaystyle
T^{**}x_{i}^{**}\perp_{B}Z^{\perp\perp}~{}\text{and~{}}x_{i}^{**}\in
M_{T^{**}}.$

(iii) Let $S\in\mathcal{L}(X^{**},Z^{\perp\perp}).$ Then using (ii), we get

$\|T^{**}-S\|\geq\|T^{**}x_{0}^{**}-Sx_{0}^{**}\|=\|T^{**}x_{0}^{**}\|=\|T^{**}\|.$

Thus, $T^{**}\perp_{B}\mathcal{L}(X^{**},Z^{\perp\perp}).$

(iv) From (ii) it follows that
$d(T^{**}x_{0}^{**},Z^{\perp\perp})=\|T^{**}x_{0}^{**}\|=\|T^{**}\|$ and from
(iii) it follows that
$d(T^{**},\mathcal{L}(X^{**},Z^{\perp\perp}))=\|T^{**}\|.$ Therefore,

$\displaystyle d(T,\mathcal{L}(X,Z))=\|T\|=\|T^{**}\|$ $\displaystyle=$
$\displaystyle d(T^{**}x_{0}^{**},Z^{\perp\perp})$ $\displaystyle=$
$\displaystyle d(T^{**},\mathcal{L}(X^{**},Z^{\perp\perp}))$
$\displaystyle\geq$ $\displaystyle\sup_{x^{**}\in
S_{X^{**}}}d(T^{**}x^{**},Z^{\perp\perp})~{}(\text{similarly~{}as~{}}(\ref{eq-
inequal}))$ $\displaystyle\geq$ $\displaystyle
d(T^{**}x_{0}^{**},Z^{\perp\perp}).$

This completes the proof. ∎

As a simple consequence of Theorem 2.2, we get the next corollary.

######  Corollary 2.3.

Let $X,Y$ be Banach spaces. Suppose $Z$ is a subspace of $Y$ such that $Z$ is
an $L^{1}-$predual space and an $M-$ideal in $Y.$ Let $T\in\mathcal{K}(X,Y).$
Suppose there exists $S\in\mathscr{L}_{\mathcal{L}(X,Z)}(T)$ such that $T-S$
is a multi-smooth operator of finite order. Then the following hold.
(i) $d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z).$
(ii) There exists $x_{0}^{**}\in S_{X^{**}}$ such that

$\displaystyle d(T^{**},\mathcal{L}(X^{**},Z^{\perp\perp}))$ $\displaystyle=$
$\displaystyle\sup_{x^{**}\in S_{X^{**}}}d(T^{**}x^{**},Z^{\perp\perp})$
$\displaystyle=$ $\displaystyle
d(T^{**}x_{0}^{**},Z^{\perp\perp})=d(T,\mathcal{L}(X,Z)).$

Additionally, if we assume that $X^{*}$ is smooth, then in (ii), we may choose
$x_{0}^{**}$ from $E_{X^{**}}.$

###### Proof.

From $S\in\mathscr{L}_{\mathcal{L}(X,Z)}(T)$ it follows that
$T-S\perp_{B}\mathcal{L}(X,Z).$ Now, using Theorem 2.2 for $T-S$, we get
$d(T-S,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx-Sx,Z).$ Since
$S\in\mathcal{L}(X,Z),$ we have
$d(T-S,\mathcal{L}(X,Z))=d(T,\mathcal{L}(X,Z))$ and $d(Tx-Sx,Z)=d(Tx,Z).$
Therefore, $d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z),$ and thus (i)
holds. Similarly, from (iv) of Theorem 2.2 we conclude that (ii) holds for
some $x_{0}^{**}\in M_{(T-S)^{**}}\subseteq S_{X^{**}}.$
If $X^{*}$ is smooth, then from (iv) of Theorem 2.2 we conclude that (ii)
holds for some $x_{0}^{**}\in M_{(T-S)^{**}}\cap E_{X^{**}}\subseteq
E_{X^{**}}.$ ∎

The following corollary shows that in Theorem 2.2, if we additionally assume
that $X$ is reflexive, then the supremum in (1) is attained.

######  Corollary 2.4.

Suppose that $X,Y,Z,T$ are as in Theorem 2.2. Moreover, assume that $X$ is
reflexive. Then for some $x_{0}\in M_{T},$ the following hold.
(i) $d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z)=d(Tx_{0},Z).$
(ii) $Tx_{0}\perp_{B}Z^{\perp\perp}.$
(iii) $T\perp_{B}\mathcal{L}(X,Z^{\perp\perp}).$

###### Proof.

From (i) and (iv) of Theorem 2.2 and the fact that $T^{**}=T$ on $X,$ we get
$x_{0}\in M_{T}$ such that

$d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z)=d(Tx_{0},Z^{\perp\perp}).$

Now, the proof follows from the observation that $Z$ is canonically embedded
in $Z^{\perp\perp}$ and therefore,

$d(Tx_{0},Z^{\perp\perp})\leq d(Tx_{0},Z)\leq\sup_{x\in S_{X}}d(Tx,Z).$

∎

Note that an important part of Theorem 2.2 depends on a special property of
the extreme points of $Z^{*},$ namely if
$\\{z_{1}^{*},z_{2}^{*},\ldots,z_{n}^{*}\\}\subseteq E_{Z^{*}}$ such that
$z_{i}^{*}\neq\pm z_{j}^{*}$ for $1\leq i\neq j\leq n,$ then the set
$\\{z_{1}^{*},z_{2}^{*},\ldots,z_{n}^{*}\\}$ is linearly independent.
Motivated by this property of the extreme points of an $L^{1}(\mu)$ space, we
define this property of a Banach space as $L^{1}-$property.

######  Definition 2.5.

Let $X$ be a Banach space. We say that $X$ has $L^{1}-$property if for any
given $\\{x_{1},x_{2},\ldots,x_{n}\\}\subseteq E_{X},$ with $x_{i}\neq\pm
x_{j}$ for $1\leq i\neq j\leq n,$ the set $\\{x_{1},x_{2},\ldots,x_{n}\\}$ is
linearly independent.

Now, we present another situation, where the minimax formula (1) is satisfied.
Most of the arguments of the following theorem are same as in Theorem 2.2. For
the sake of convenience, we give a sketch of the proof here.

######  Theorem 2.6.

Suppose $X$ is a Banach space such that $X^{**}$ satisfies $L^{1}-$property.
Let $Y$ be an arbitrary Banach space and $Z$ be a closed subspace of $Y.$ Let
$T\in\mathcal{K}(X,Y)$ be a multi-smooth operator of finite order. Suppose
that $T\perp_{B}\mathcal{L}(X,Z).$ Then the conditions (i)-(iv) of Theorem 2.2
hold. Moreover, in this case we may choose $x_{0}^{**}$ of (ii) and (iv) from
$E_{X^{**}}\cap M_{T^{**}}.$

###### Proof.

As in Theorem 2.2, we get $\lambda_{i}>0,$ $x_{i}^{**}\in
E_{X^{**}},y_{i}^{*}\in E_{Y^{*}}\cap M_{T^{*}}$ for $1\leq i\leq h$ such that
$\sum_{i=1}^{h}\lambda_{i}=1,$ $x_{i}^{**}(T^{*}y_{i}^{*})=\|T\|$ and
$\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}(A^{*}y_{i}^{*})=0$ for all
$A\in\mathcal{L}(X,Z).$ Since $X^{**}$ satisfies $L^{1}-$property, without
loss of generality, we may assume that $\\{x_{1}^{**},\ldots,x_{h}^{**}\\}$ is
linearly independent. Choose
$x^{*}\in\cap_{i=2}^{h}\ker(x_{i}^{**})\setminus\ker(x_{1}^{**}).$ Let $z\in
Z$ be arbitrary. Define $A\in\mathcal{L}(X,Z)$ by$A(x)=x^{*}(x)z$ for all
$x\in X.$ Then from $\sum_{i=1}^{h}\lambda_{i}x_{i}^{**}(A^{*}y_{i}^{*})=0,$
we get $\lambda_{1}x_{1}^{**}(x^{*})y_{1}^{*}(z)=0.$ Thus, $y_{1}^{*}(z)=0.$
Therefore, $y_{1}^{*}\in Z^{\perp}.$ The rest of the proof follows proceeding
similarly as in Theorem 2.2. ∎

######  Remark 2.7.

Recall from [9] that a Banach space $W$ is an $L^{1}-$predual space if and
only if $W^{**}$ is isometrically isomorphic to $C(K)$ for some extremally
disconnected compact Hausdorff space $K.$ Thus, $W^{**}$ is again an
$L^{1}-$predual space. Hence, in the Theorem 2.6 we may consider $X=W^{*},$
where $W$ is an $L^{1}-$predual space. Suppose that, $K$ is a Hyperstonian
space and $N(K,\mathbb{R})^{+}$ is set of all positive normal regular Borel
measures on $K.$ Let
$N(K,\mathbb{R})=N(K,\mathbb{R})^{+}-N(K,\mathbb{R})^{+}.$ Then from Theorem
[9, Th. 10, pp 95], it follows that $N(K,\mathbb{R})^{*}$ is isometrically
isomorphic to $C(K),$ which is an $L^{1}-$predual space. Therefore, Theorem
2.6 in particular holds for $X=N(K,\mathbb{R}).$ In other words, if $X$ is an
abstract $L_{1}-$space, then $X^{*}$ is an abstract $M-$space and $X^{*}=C(K)$
for some compact Hausdorff space $K$ (see [9, pp. 97]). Since $C(K)$ is an
$L_{1}-$predual space, Theorem 2.6 holds for an abstract $L_{1}-$ space $X.$
For the definition of abstract $L_{1}-$space, abstract $M-$space and related
results see [9].

######  Remark 2.8.

Note that from Theorem 2.2 (Theorem 2.6), we get the following implications

$T\perp_{B}\mathcal{L}(X,Z)\Rightarrow
T^{**}x_{0}^{**}\perp_{B}Z^{\perp\perp}\Rightarrow
T^{**}\perp_{B}\mathcal{L}(X^{**},Z^{\perp\perp}),$

for some $x_{0}^{**}\in M_{T^{**}}.$ Recall from [18, Prop. 1.11.14, pp 102]
that $Z^{**}$ is isometrically isomorphic to $Z^{\perp\perp}.$ Therefore, the
space $\mathscr{A}=\\{S^{**}:S\in\mathcal{L}(X,Z)\\}$ is a subspace of
$\mathcal{L}(X^{**},Z^{\perp\perp}).$ Thus,
$T^{**}\perp_{B}\mathcal{L}(X^{**},Z^{\perp\perp})$ yields that
$T^{**}\perp_{B}\mathscr{A},$ i.e., $T^{**}\perp_{B}S^{**}$ for all
$S\in\mathcal{L}(X,Z).$ Now, from the equality $\|T-S\|=\|T^{**}-S^{**}\|,$ we
get that $T\perp_{B}\mathcal{L}(X,Z).$ Therefore, if $X,Y,Z$ and $T$ satisfy
the conditions of Theorem 2.2 (respectively, Theorem 2.6), then we get the
following equivalence:

$T\perp_{B}\mathcal{L}(X,Z)\Leftrightarrow
T^{**}x_{0}^{**}\perp_{B}Z^{\perp\perp}\Leftrightarrow
T^{**}\perp_{B}\mathcal{L}(X^{**},Z^{\perp\perp}),$

for some $x_{0}^{**}\in M_{T^{**}}$ (respectively, $x_{0}^{**}\in
M_{T^{**}}\cap E_{X^{**}}$).

In [19, Prop. 2], Rao proved that if the supremum in (1) is attained, then
$d(T^{**},\mathcal{L}(X^{**},Z^{\perp\perp}))=\sup_{x^{**}\in
S_{X^{**}}}d(T^{**}x^{**},Z^{\perp\perp})=d(T,\mathcal{L}(X,Z)).$ Note that,
Theorem 2.2 and Theorem 2.6 provide us other situations, where such equality
holds even if the supremum in (1) is not attained.
Next, we show that if $X$ is reflexive and $Z$ is an arbitrary closed subspace
of $Y,$ then some restriction on the norm attainment set of $T$ yields (1).

######  Theorem 2.9.

Let $X$ be a reflexive Banach space and $Y$ be an arbitrary Banach space.
Suppose $Z$ is a closed subspace of $Y.$ Let $T\in\mathcal{K}(X,Y)$ be a
multi-smooth operator of finite order and $M_{T}\cap E_{X}=\\{\pm x_{i}\in
S_{X}:1\leq i\leq n\\},$ where $\\{x_{1},x_{2},\ldots,x_{n}\\}$ is linearly
independent. Suppose that $T\perp_{B}\mathcal{L}(X,Z).$ Then there exists
$x_{0}\in M_{T}\cap E_{X}$ such that the following hold.
(i) $d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z)=d(Tx_{0},Z).$
(ii) $Tx_{0}\perp_{B}Z^{\perp\perp}.$
(iii) $T\perp_{B}\mathcal{L}(X,Z^{\perp\perp}).$

###### Proof.

Since $T$ is a multi-smooth operator of finite order, following similar
arguments as Theorem 2.2, we get scalars
$\lambda_{1},\lambda_{2},\ldots,\lambda_{h}>0$ and
$f_{1},f_{2},\ldots,f_{h}\in J(T)\cap E_{\mathcal{K}(X,Y)^{*}}$ such that
$\sum_{i=1}^{h}\lambda_{i}=1$ and $\sum_{i=1}^{h}\lambda_{i}f_{i}(A)=0$ for
all $A\in\mathcal{L}(X,Z).$ Since $X$ is reflexive, from (6) it follows that

$E_{\mathcal{K}(X,Y)^{*}}=\\{y^{*}\otimes x:y^{*}\in E_{Y^{*}},x\in E_{X}\\},$

where $y^{*}\otimes x(S)=y^{*}(Sx)$ for each $S\in\mathcal{L}(X,Y).$
Therefore, for $1\leq i\leq h,$ $f_{i}=y_{i}^{*}\otimes x_{i},$ where
$y_{i}^{*}\in E_{Y^{*}},x_{i}\in E_{X}.$ Now, $y_{i}^{*}\otimes x_{i}\in J(T)$
implies that $x_{i}\in M_{T}$ and $y_{i}^{*}\in J(Tx_{i}).$ Without loss of
generality, we may assume that $\\{x_{i}:1\leq i\leq h\\}$ is linearly
independent. Now, choose $x^{*}\in X^{*}$ such that $x^{*}(x_{i})=0$ for all
$2\leq i\leq h$ and $x^{*}(x_{1})\neq 0.$ Choose an arbitrary $z\in Z.$
Consider $A\in\mathcal{L}(X,Z)$ defined as $Ax=x^{*}(x)z$ for all $x\in X.$
Then

$\displaystyle\sum_{i=1}^{h}\lambda_{i}f_{i}(A)=0$ $\displaystyle\Rightarrow$
$\displaystyle\sum_{i=1}^{h}\lambda_{i}y_{i}^{*}\otimes x_{i}(A)=0$
$\displaystyle\Rightarrow$
$\displaystyle\sum_{i=1}^{h}\lambda_{i}y_{i}^{*}(Ax_{i})=0$
$\displaystyle\Rightarrow$
$\displaystyle\sum_{i=1}^{h}\lambda_{i}y_{i}^{*}(z)x^{*}(x_{i})=0$
$\displaystyle\Rightarrow$
$\displaystyle\lambda_{1}y_{1}^{*}(z)x^{*}(x_{1})=0\Rightarrow
y_{1}^{*}(z)=0.$

Since $z\in Z$ is chosen arbitrarily, we get $y_{1}^{*}\in Z^{\perp}.$ Since
for each $u\in Z^{\perp\perp},$ $u(y^{*})=0$ holds, we get

$\|Tx_{1}+u\|\geq|(Tx_{1}+u)y_{1}^{*}|=|y_{1}^{*}(Tx_{1})|=\|Tx_{1}\|.$

Thus, $Tx_{1}\perp_{B}Z^{\perp\perp}.$ This proves (ii). Moreover,
$Tx_{1}\perp_{B}Z^{\perp\perp}$ implies that $Tx_{1}\perp_{B}Z,$ since $Z$ is
canonically embedded in $Z^{\perp\perp}.$ Now, (i) follows from the following
inequality:

$\displaystyle
d(Tx_{1},Z)=\|Tx_{1}\|=\|T\|=d(T,\mathcal{L}(X,Z))\geq\sup_{x\in
S_{X}}d(Tx,Z)\geq d(Tx_{1},Z),$

where the first inequality follows from (2).
To prove (iii), let $S\in\mathcal{L}(X,Z^{\perp\perp}).$ Observe that

$\|T-S\|\geq\|Tx_{1}-Sx_{1}\|\geq\|Tx_{1}\|=\|T\|,$

where the second inequality follows from $Tx_{1}\perp_{B}Z^{\perp\perp}.$ Thus
$T\perp_{B}\mathcal{L}(X,Z^{\perp\perp}).$ This completes the proof of the
theorem. ∎

We immediately get the next corollary due to Theorem 2.9.

######  Corollary 2.10.

Suppose $X=\ell_{1}^{n},$ $Y$ is a Banach space and $Z$ is a closed subspace
of $Y.$ Assume that each non-zero element of $Y$ is a multi-smooth point of
finite order. Let $T\in\mathcal{L}(X,Y)\setminus\mathcal{L}(X,Z)$ be such that
$\mathscr{L}_{\mathcal{L}(X,Z)}(T)\neq\emptyset.$ Then there exists $x_{0}\in
E_{X}$ such that

$d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z)=d(Tx_{0},Z).$

In particular, if $T\perp_{B}\mathcal{L}(X,Z)$ then
$Tx_{0}\perp_{B}Z^{\perp\perp},$ for some $x_{0}\in M_{T}\cap E_{X}.$

###### Proof.

Suppose that $(0\neq)~{}S\in\mathcal{L}(X,Y).$ Since $X=\ell_{1}^{n},$
$M_{S}\cap E_{X}$ is of the form $\\{\pm x_{i}:1\leq i\leq h\\},$ where
$\\{x_{i}:1\leq i\leq h\\}$ is linearly independent. Moreover, $Sx_{i}\in Y$
is a multi-smooth point of finite order for each $i.$ Therefore, using [16,
Cor. 2.3] we get $S$ is a multi-smooth operator of finite order. Thus, each
non-zero operator of $\mathcal{L}(X,Y)$ satisfies the hypothesis of Theorem
2.9. Now, suppose that $S_{0}\in\mathscr{L}_{\mathcal{L}(X,Z)}(T).$ Then
$T-S_{0}\perp_{B}\mathcal{L}(X,Z).$ Note that, since
$T\notin\mathcal{L}(X,Z),T-S_{0}\neq 0.$ Now, using Theorem 2.9, we get
$x_{0}\in E_{X}$ such that

$\displaystyle d(T-S_{0},\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx-
S_{0}x,Z)=d(Tx_{0}-S_{0}x_{0},Z)$ $\displaystyle\Rightarrow$ $\displaystyle
d(T,\mathcal{L}(X,Z))=\sup_{x\in S_{X}}d(Tx,Z)=d(Tx_{0},Z).$

On the other hand, if $T\perp_{B}\mathcal{L}(X,Z),$ then from (ii) of Theorem
2.9, it follows that $Tx_{0}\perp_{B}Z^{\perp\perp}$ for some $x_{0}\in
M_{T}\cap E_{X}.$ This completes the proof. ∎

Note that, in particular, Corollary 2.10 holds for a smooth Banach space $Y.$
Furthermore, from [12, Cor. 2.2], it follows that every nonzero operator of
$\mathcal{K}(H_{1},H_{2})$ is a multi-smooth operator of finite-order, where
$H_{1},H_{2}$ are Hilbert spaces. For other examples of Banach spaces $Y,$
where each non-zero element is a multi-smooth point of finite order see [8,
11]. As a consequence of the results obtained here, we get the following
corollary providing sufficient condition for proximinality of a subspace.

######  Corollary 2.11.

Suppose $X=\ell_{1}^{n},$ $Y$ is a Banach space and $Z$ is a closed subspace
of $Y.$ Assume that each non-zero element of $Y$ is a multi-smooth point of
finite order. Suppose that $\mathcal{L}(X,Z)$ is a proximinal subspace of
$\mathcal{L}(X,Y).$ Then $Z$ is a proximinal subspace of $Y.$

###### Proof.

Let $y\in Y\setminus Z.$ Choose $x^{*}\in E_{X^{*}}.$ Let
$T\in\mathcal{L}(X,Y)\setminus\mathcal{L}(X,Z)$ be defined as $Tx=x^{*}(x)y$
for all $x\in X.$ From the proximinality of $\mathcal{L}(X,Z)$ in
$\mathcal{L}(X,Y),$ it follows that
$\mathscr{L}_{\mathcal{L}(X,Z)}(T)\neq\emptyset.$ Let
$S\in\mathscr{L}_{\mathcal{L}(X,Z)}(T).$ Then $T-S\perp_{B}\mathcal{L}(X,Z).$
Therefore, using Corollary 2.10, we get $Tx_{0}-Sx_{0}\perp_{B}Z$ for some
$x_{0}\in E_{X}.$ Observe that $|x^{*}(x_{0})|=1,$ since $X=\ell_{1}^{n}.$
Thus, $x^{*}(x_{0})y-Sx_{0}\perp_{B}Z,$ which yields that
$\frac{1}{x^{*}(x_{0})}Sx_{0}\in\mathscr{L}_{Z}(y).$ This completes the proof.
∎

We would like to end the paper with the remark that Corollary 2.11 is
motivated from [19, Cor. 5]. However, in [19, Cor. 5], one of the assumptions
is that $Z$ is an $M-$ideal in $Y,$ which itself is a sufficient condition for
the proximinality of $Z$ in $Y$ (see [6, Prop. 1.1, pp 50]). Here we emphasize
that in Corollary 2.11, $Z$ is assumed to be an arbitrary closed subspace of
$Y.$

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