Subdifferential set of an operator

Abstract

Let X, Y be complex Banach spaces. Let ${\mathcal L}(X,Y)$ be the space of bounded operators. An important aspect of understanding differentiability is to study the subdifferential of the norm at a point, say $x \in X$, this is the set, $\{f \in X^*:\Vert f\Vert =1~,f(x)=\Vert x\Vert \}$. See page 7 in Deville et al. (Pitman Monographs and Surveys in Pure and Applied Mathematics. 64. Harlow: Longman Scientific and Technical. New York: John Wiley and Sons, Inc. 1993). Motivated by recent results of Singla (Singla in Linear Alg. Appl. 629:208–218, 2021) in the context of Hilbert spaces, for $T \in {{\mathcal {L}}}(X,Y)$, we determine the subdifferential of the operator norm at T, $\partial _T = \{\Lambda \in {{\mathcal {L}}}(X,Y)^*: \Lambda (T) = \Vert T\Vert ~,~\Vert \Lambda \Vert =1\}$. Our approach is based on the ‘position’ of the space of compact operators and the Calkin norm of T. Our ideas give a unified approach and extend several results from Singla (Linear Alg. Appl. 629:208–218, 2021) to the case of $\ell ^p$-spaces for $1<p<\infty$. We also investigate the converse, using the structure of the subdifferential set to decide when the Calkin norm is a strict contraction. As an application of these ideas, we partially solve the open problem of relating the subdifferential of the operator norm at a compact operator T to that of $T(x_0)$, where $x_0$ is a unit vector where T attains its norm.

1 Introduction

Let X, Y be complex Banach spaces. An interesting problem in operator theory is to determine standard Banach space theoretic sets in ${{\mathcal {L}}}(X,Y)$, based on geometric properties of X, Y and their duals.

Let $X_1$ denote the closed unit ball and let $\partial _e X^*_1$ be the set of extreme points of the dual unit ball. In the context of dual spaces, closure operation is always with respect to the weak$^*$ -topology. For notational simplicity, we call the subdifferential of the norm at a vector x, $\partial _x =\{f \in X^*: \Vert f\Vert =1,~f(x)=\Vert x\Vert \}$, as the subdifferential set, with the vector under consideration appearing in the subscript . This notation is consistent with [2]. Study of these sets in various categories of Banach spaces has long history (an extensive bibliography is available in [3]) and we only quote the results of Taylor and Werner (see [4] and [5]) to illustrate.

For $y \in X$, the relation to the directional derivative of the norm at x is given by the formula (see [6] and [1], pages 7,8)

$$\begin{aligned} lim_{t\rightarrow 0^+} \frac{\Vert x+ty\Vert -\Vert x\Vert }{t} = sup \{re(f(y)): f \in \partial _x\}. \end{aligned}$$

We recall that $\partial _x$ is an extreme, weak$^*$-closed convex subset of $X^*_1$ ( called a face, see [7] for basic convexity theory) and therefore the set of extreme points, $\partial _e (\partial _x)= \partial _x \cap \partial _e X^*_1$. Thus we can replace the functionals in the formula by extreme points. The idea now is to replace the functionals in the RHS set by those coming from an appropriate subspace of X, where the extreme functionals have specific description. Note that the limit on LHS is independent of this manipulation, which is the prime motivation of this approach.

For $x^{**} \in X^{**}$, $y^*\in Y^*$, $x^{**} \otimes y^*$ denotes the functional $(x^{**} \otimes y^*)(T) = x^{**}(T^*(y^*))$. We recall from [8] (Theorem VI.1.3) that for any subspace J of compact operators ${{\mathcal {K}}}(X,Y)$ , $\partial _eJ^*_1 \subset \{x^{**} \otimes y^*: x^{**} \in \partial _e X^{**}_1,~y^*\in \partial _e Y^*_1\}$ and equality holds if J also contains all finite rank operators. For $A \in {\mathcal {L}}(X,Y)$, such that $d(A, {{\mathcal {K}}}(X,Y))<1$, our main result is to determine $\partial _A$ depending on the position of the space of compact operators.

We recall from Chapter I of [8], that a closed subspace $J \subset X$ is said to be a M-ideal, if there is a linear projection $P: X^*\rightarrow X^*$ such that $ker(P) = J^\bot$ and $\Vert x^*\Vert = \Vert P(x^*)\Vert +\Vert x^*-P(x^*)\Vert$ for all $x^*\in X^*$. When this happens, functionals in $J^*$ have unique norm-preserving extension in $X^*$ and $J^*$ is canonically identified with the range of P so that $X^*= J^\bot \bigoplus _1 J^*$, consequently $\partial _e X^*_1 = \partial _e J^*_1 \cup \partial _e J^\bot _1$.

See [8] for examples of M-ideals from function spaces, Banach algebras and spaces of operators. In particular Chapter VI deals with examples where ${{\mathcal {K}}}(X,Y)$ is a M-ideal in ${{\mathcal {L}}}(X,Y)$, which includes, ${{\mathcal {K}}}(\ell ^p)$ in ${{\mathcal {L}}}(\ell ^p)$ for $1<p<\infty$ and ${{\mathcal {K}}}(X,c_0)$ is a M-ideal in ${{\mathcal {L}}}(X,c_0)$ as well as $\bigoplus _{\infty } X$, for any Banach space X. Note also M-ideals in $C^*$-algebras are precisely closed two-sided ideals. This explains the overlap with [2] and gives possible extensions of the results there beyond $C^*$-algebras. See also [3].

We investigate conditions on X, Y which will allow us to determine the Calkin norm (i.e., the quotient norm of ${\mathcal L}(X,Y)/{{\mathcal {K}}}(X,Y)$) of an operator, knowing the behaviour of the subdifferential set. We show that under the general conditions assumed here, if 0 is not a weak$^*$ accumulation point of $\partial _e X^{**}_1~,~\partial _e Y^*_1$, then the Calkin norm indeed gets determined.

Since the subdifferential set $\partial _x$ can be used to evaluate the limit, for $y \in X$, $lim_{t\rightarrow 0^+} \frac{\Vert x+ty\Vert -\Vert x\Vert }{t}$, a natural question is, when X is canonically embedded in its bidual $X^{**}$, if one can replace y in the above formula, by $\tau \in X^{**}$, while retaining $\partial _x$ in the RHS? We partially answer this, using the notion of point of norm-weak upper semi-continuity of the preduality map on $X^{**}$ (see the preamble before Proposition 11).

We also show that for the special classes of compact operators ${{\mathcal {K}}}(X,Y)$, when they form a M-ideal in ${\mathcal {L}} (X,Y)$ and ${{\mathcal {L}}}(X,Y)$ is the bidual of ${\mathcal {K}} (X,Y)$ (for example, when X and Y are $\ell ^p$-spaces, $1< p <\infty$), the subdifferential continues to behave well in all higher even order duals of ${{\mathcal {K}}}(X,Y)$.

Another interesting problem is to relate the subdifferential of the operator norm at T to that of $T(x_0)$, where $x_0$ is a point where T attains its norm. Using some recent work from [9], we give conditions when this can be solved for a $T \in {\mathcal {K}} (X,Y)$, for a reflexive space X, $J \subset Y$ is a M-ideal.

For a compact Hausdorff space $\Omega$, let $C(\Omega ,Y)$ be the space of Y-valued continuous functions, equipped with the supremum norm. For $\omega \in \Omega$, $y^*\in Y^*$, $\delta (\omega )\otimes y^*$ denotes the functional $(\delta (\omega ) \otimes y^*)(f)=y^*(f(w))$. For a M-ideal $J \subset Y$ and $f \in C(\Omega ,Y)$, if $\partial _f =\overline{CO}\{\delta (\omega )\otimes y^*:y^*(f(\omega ))=\Vert f\Vert ,~\omega \in \Omega ~,y^*\in \partial _e J^*_1\}$, (here and elsewhere, CO stands for the convex hull of the set in $\{.\}$), when 0 is not a weak$^*$-closed accumulation point of $\partial _e Y^*_1$, there is a $\omega _0 \in \Omega$ such that $\Vert f(\omega _0)\Vert =\Vert f\Vert$ and $\partial _{f(\omega _0)}=\overline{CO}\{y^*\in \partial _e J^*_1: y^*(f(\omega _0))= \Vert f\Vert \}$.

2 Geometry of the subdifferential set

We first give a general form of Theorem 1.4 from [2] (see also the Zbl review by this author). Let $J \subset X$ be a closed subspace. We note that if $x \in J$, then relative to J, $\partial _x = \overline{CO} \{f \in \partial _e J^*_1: f(x)=1\}$, where the closure is in the weak$^*$-topology of $J^*$.

Theorem 1

Let $J \subset X$ be a M-ideal. Let $x \in X$ be a unit vector such that $d(x,J)<1$. Then $\partial _x = \overline{CO}\{f \in \partial _e J^*_1: f(x) = 1\}$. Here the closure is taken in the weak$^*$-topology of $X^*$. In particular $\partial _x$ has the same description as when $x \in J$. Thus for $y \in X$,

$$\begin{aligned} lim_{t\rightarrow 0^+} \frac{\Vert x+ty\Vert -\Vert x\Vert }{t} = sup \{re(f(y)): f \in \partial _e J^*_1,~f(x)=1\}. \end{aligned}$$

Proof

Since J is a M-ideal, we have, $X^*= J^*\bigoplus _1 J^\bot$. Thus if the set $\{f \in \partial _eJ^*_1: f(x) = 1\}$ is non-empty, then it is contained in $\partial _x$.

Since $\partial _x$ is a weak$^*$-closed extreme convex subset of $X^*_1$, we will show that any extreme point $g \in \partial _x$ is in $J^*$ . This completes the proof.

Since $g \in \partial _e X^*_1$, in view of decomposition of $X^*$, suppose $g \in J^\bot$. Now $g(x) = 1$ implies $d(x,J) = 1$. Therefore, $g \in \partial _eJ^*_1$. $\square$

Remark 2

Since y is weak$^*$-continuous on $J^*$, when $\partial _e J^*_1$ is weak$^*$-closed in the weak$^*$-topology of $J^*$, we have

$$\begin{aligned} lim_{t\rightarrow 0^+} \frac{\Vert x+ty\Vert -\Vert x\Vert }{t} = re(f_0(y)) \end{aligned}$$

for some $f_0 \in \partial _e J^*_1$, $f_0(x)=1$. This in particular always happens (under the above hypothesis) when $\partial _e X^*_1$ is weak$^*$-closed.

The following corollary follows immediately from our specific knowledge of extreme points of ${{\mathcal {K}}}(X,Y)^*_1$.

Corollary 3

Let X, Y be Banach spaces and let $J \subset {{\mathcal {K}}}(X,Y) \subset {{\mathcal {L}}}(X,Y)$ be a M-ideal in ${{\mathcal {L}}}(X,Y)$. For any $A \in {{\mathcal {L}}}(X,Y)$ such that $d(A,J)<1$, $\partial _A = \overline{CO} \{x^{**} \otimes y^*\in \partial _eJ^*_1: x^{**}(A^*(y^*))=1,~x^{**} \in \partial _e X^{**}_1~,y^*\in \partial _e Y^*_1\}$.

Remark 4

In particular we have $A^*$ attains its norm and hence so are all the adjoints of higher order of A. More generally we note that if a $B \in {{\mathcal {L}}}(X,Y)$, is such that $B^*$ attains its norm, then it attains it at an extreme point $y^*\in Y^*_1$ (see [10]). Since $\Vert B\Vert = \Vert B^*(y^*)\Vert$, let $x^{**} \in \partial _e X^{**}_1$ be such that $x^{**}(B^*(y^*))= (x^{**} \otimes y^*)(B) = \Vert B\Vert$. Now if the J above contains all finite rank operators, $x^{**} \otimes y^*\in \partial _e J^*_1$. It may be worth recalling here that the set of operators whose adjoint attains its norm is dense in ${{\mathcal {L}}}(X,Y)$, see [11]. When $X=Y= \ell ^p$ for $1<p<\infty$, $J ={{\mathcal {K}}}(\ell ^p)$, we have for $A \in {{\mathcal {L}}}(\ell ^p)$ with $d(A,{{\mathcal {K}}}(\ell ^p))<\Vert A\Vert$, $\partial _A = \overline{CO} \{x \otimes y: A(x)(y)= \Vert A\Vert ,~x \in \ell ^p_1 ~,y \in \ell ^q_1\}$, thus extending Theorem 1.4 from [2].

Let $Z \subset Y$ be a closed subspace. The following proposition identifies subdifferential limits of vectors in X/Z. In what follows, by $\pi : X \rightarrow X/Z$ we denote the quotient map.

Proposition 5

Let $Z \subset Y \subset X$ be Banach spaces. Suppose Y is a M-ideal in X and $x \in X$ is such that $d(x,Y)<d(x,Z)$. Then for $x'\in X$,

$$\begin{aligned}&lim_{t \rightarrow 0^+} \frac{\Vert \pi (x)+t\pi (x')\Vert -\Vert \pi (x)\Vert }{t}\\&\quad = sup\{re(f(x')): f \in \partial _e (Y^*\cap Z^\bot )_1~,~f(x)=1\}. \end{aligned}$$

Proof

By Proposition I.1.7 in [8], we have that Y/Z is a M-ideal in X/Z. Hence $Z^\bot = Y^\bot \bigoplus _1 (Y^*\cap Z^\bot )$. For any $y \in Y$, $\Vert \pi (x)-\pi (y)\Vert =\Vert \pi (x-y)\Vert \le \Vert x-y\Vert$. Thus $d(\pi (x),Y/Z) \le d(x,Y)<d(x,Z)=\Vert \pi (x)\Vert$. Now identifying $(Y/Z)^*= Y^*\cap Z^\bot \subset (X/Z)^*$, the conclusion follows from Theorem 1. $\square$

We next consider the question, suppose J is a M-ideal in X and the formula for $\partial _x$ is valid, when is $d(x,J)<1?$. The following example illustrates how it can depend on the choice of the M-ideal.

Example 6

Consider $c_0 \subset \ell ^\infty$, a well known example of a M-ideal (algebraic ideal) of $\ell ^\infty$.  Let $x \in \ell ^\infty$ be a unit vector, with all but finitely many coordinates coming from the unit circle $\Gamma$, the rest from the open disc. Using the identification of $\ell ^\infty = C(\beta (\mathbf{N}))$, we see that $\partial _e(\partial _x) \subset \Gamma \{\delta (\tau ): \tau \in \beta (\mathbf{N})\}$. Thus we get $\partial _x = \overline{CO}\{\overline{t}\delta (k): k\in \mathbf{N}~, t=x(k)~,|x(k)|=1\}$, where $\delta (\tau )$ is the Dirac measure. Clearly $\delta (k) \in \partial _e (\ell ^1 = c_0^*)_1$. Now for the ideal $J = \{\alpha \in \ell ^\infty : \alpha (k) = 0~for~k~with ~|x(k)|<1 \})$, J determines $\partial _x$, as well as $c_0$. However $d(x, c_0) = 1$ and $d(x,J)<1$.

We give two positive answers depending on the weak$^*$-closure of $\partial _e J^*_1$ in $J^*$ and $X^*$.

Theorem 7

Suppose for X, Y, J as in Theorem 1, Suppose $\partial _x = \overline{CO}\{f \in \partial _e J^*_1: f(x) = 1\}$.

• (1) Suppose $0 \notin \overline{\partial _e J^*_1}$, closure in the weak$^*$-topology of $J^*$.

• (2) Suppose $\overline{\partial _e J^*_1} \subset [0,1]\partial _e J^*_1$, closure taken on the weak$^*$-topology of $X^*$.

In either case, $d(x,J) <1$.

Proof

To see this assuming 1), if $d(x,J) = 1$, choose $\tau \in J^\bot _1 \cap \partial _e X^*$ such that $\tau (x) = 1$ . Since $\tau \in \partial _x$ by Milman’s converse of the Krein-Milman theorem, $\tau \in \overline{\{f \in \partial _e J^*_1: f(x) = 1\}}$, closure in the weak$^*$-topology of $X^*$. Let $\{f_{\alpha }\}$ be a net from the set in RHS such that $f_{\alpha } \rightarrow \tau$ in the weak$^*$-topology of $X^*$. Since $\tau (J) = 0$, we get a contradiction. So $d(x,J) <1$.

Now suppose we have 2). Since $\tau \in \overline{\partial _e J^*_1}$, for the net $\{f_{\alpha }\}$ as in the proof of 1), we get by hypothesis in 2), $\tau = \lambda h$ for $h \in \partial _e J^*_1$, $\lambda \in [0,1]$. Since $\Vert \tau \Vert =1=\Vert h\Vert$, we get $\lambda =1$. Hence our claim follows as before. $\square$

Remark 8

Examples of spaces that satisfy above extremal conditions include spaces of the form $\{f \in C(\Omega ): f(t_{\alpha })=\lambda _{\alpha } f(s_{\alpha })~for~all~\alpha \in \Delta \}$. Here $\Omega$ is a compact Hausdorff space, $\Delta$ is any index set,

$$\begin{aligned} \{t_{\alpha }\}_{\alpha \in \Delta }~,~\{s_{\alpha }\}_{\alpha \in \Delta } \subset \Omega ~ and ~\{\lambda _{\alpha }\}_{\alpha \in \Delta } \subset [-1,1]. \end{aligned}$$

We next consider conditions specific to spaces of compact operators. We recall that in the case of $\ell ^p$-spaces, 0 is a weak-accumulation point of the unit sphere.

Proposition 9

Suppose $J\subset {{\mathcal {K}}}(X,Y) \subset {{\mathcal {L}}}(X,Y)$ is a M-ideal in ${{\mathcal {L}}}(X,Y)$ and it contains all finite rank operators. Assume further, 0 is not a weak$^*$ extreme point of $\partial _e X^{**}_1$ and $\partial _e Y^*_1$. Let $A \in {{\mathcal {L}}}(X,Y),~\Vert A\Vert =1$ and $\partial _A = \overline{CO} \{x^{**} \otimes y^*: x^{**}(A^*(y^*))=1~,~x^{**} \in \partial _e X^{**}_1,~y^*\in \partial _eY^*_1\}$. Then $d(A,J) < 1$.

Proof

We will show that $0 \notin \overline{\partial _e J^*_1}$. If not, in view of the description of the extreme points, we get two nets indexed by the same set, $\{x^{**}_{\alpha }\} \subset \partial _e X^{**}_1$ and $\{y^*_{\alpha }\} \subset \partial _e Y^*_1$ such that the functionals $x^{**}_{\alpha } \otimes y^*_{\alpha } \rightarrow 0$ in the weak$^*$-topology of $J^*$. Using weak$^*$-compactness, by going through appropriate subnets, if necessary, we may and do assume that $x^{**}_{\alpha } \rightarrow x^{**}$ and $y^*_{\alpha } \rightarrow y^*$ in the weak$^*$-topology, for some $x^{**} \ne 0 \ne y^*$. For any $T \in J$, since T is compact, $T^*(y^*_{\alpha }) \rightarrow T^*(y^*)$ in the norm. Consequently $x^{**}_{\alpha }(T^*(y^*_{\alpha })) \rightarrow x^{**}(T^*(y^*))$. Therefore $x^{**}_{\alpha } \otimes y^*_{\alpha } \rightarrow x^{**}\otimes y^*$ in the weak$^*$-topology of $J^*$. So $x^{**} \otimes y^*= 0$ on J . A contradiction since J contains all operators of rank one. $\square$

We next apply this to the situation when X is a M-ideal in the canonical embedding, in its bidual $X^{**}$, the so called M-embedded spaces. See Chapter III of [8], for examples among spaces of operators, particularly when for reflexive Banach spaces X, Y ${{\mathcal {L}}}(X,Y)$ is the canonical bidual of ${{\mathcal {K}}}(X,Y)$ (like in the case of Hilbert spaces and $\ell ^p$-spaces).

Proposition 10

Suppose X is a M-ideal in $X^{**}$. Let $\tau \in X^{**}$ be such that $d(\tau ,X) < \Vert \tau \Vert$, then $\tau$ attains its norm. The set $\partial \tau$ has the same description in all higher odd ordered duals of X.

Proof

Since X is a M-ideal in $X^{**}$, the canonical projection $\Lambda \rightarrow \Lambda |X$ is a L-projection, thus $X^{***} = X^{*} \bigoplus _1 X^\bot$ for the canonical embedding of $X^*$ in $X^{***}$. We see that the hypothesis implies there is a $g \in \partial _e X^*_1$ such that $\tau (g) =1$. The main result of [12] says that under the hypothesis, X continues to be a M-ideal of all higher even ordered duals of X. Hence the conclusion follows. $\square$

Let X be canonically embedded in its bidual, $X^{**}$. Another interesting question is to consider when the equality,

$$\begin{aligned} lim_{t\rightarrow 0^+} \frac{\Vert x+t\tau \Vert -\Vert x\Vert }{t} = sup \{re(\tau (f)): f \in \partial _x\} \end{aligned}$$

holds for any $\tau \in X^{**}$.

Our next proposition answers this question using the notion of the preduality mapping on $X^{**}$ (see [13]). For $x \in X$ considered as an element of $X^{**}$, we let $\partial '_x = \{\tau \in X^{***}_1: \tau (x) = \Vert \tau \Vert \}$.

We recall that for a norm attaining functional $\Lambda \in X^{**}$, attaining its norm at $f_0 \in X^{*}_1$ is a point of norm-weak upper semi-continuity for the preduality map, if for any weak neighbourhood V of 0 in $X^*$, there is a $\delta >0$ such that for any $\Lambda ' \in X^{**}$, $\Vert \Lambda '-\Lambda \Vert < \delta$, $\{g \in X^*_1:\Lambda '(g) = \Vert \Lambda '\Vert \} \subset \{h \in X^*_1: \Lambda (h) = \Vert \Lambda \Vert \}+V$. Part of our next proposition is perhaps folklore, as we are unable to find a reference, we include its easy proof. It is well known that any unitary in a $C^*$- algebra satisfies the condition assumed in the following proposition.

Proposition 11

Let $x \in X$ be a point of norm-weak upper semi-continuity for the preduality mapping of $X^{**}$. Then

$$\begin{aligned} lim_{t\rightarrow 0^+} \frac{\Vert x+t\tau \Vert -\Vert x\Vert }{t} = sup \{re(\tau (f)): f \in \partial _x\} \end{aligned}$$

for any $\tau \in X^{**}$.

Proof

It follows from Lemma 2.2 in [13] that the hypothesis implies $\partial _x$ is weak$^*$-dense (in the weak$^*$-topology of $X^{***}$) in $\partial '_x$. Since $\tau$ is weak$^{*}$-continuous on $X^{***}$, the conclusion follows. $\square$

Remark 12

We note that if $J \subset {X}$ is a M-ideal, then J need not be a M-ideal in $X^{**}$ ( it is easy to see that in C([0, 1]), the ideal $J = \{f \in C([0,1]): f([0,\frac{1}{2}])=0\}$ is not an ideal in $C([0,1])^{**}$).

3 Application

As an application of the previous set of ideas, we formulate the compact operator version of the operator norm subdifferential problem and provide a partial solution.

Let X be a reflexive Banach spaces and suppose Y has the metric approximation property. Suppose $J \subset Y$ be a M-ideal and let $T \in {{\mathcal {K}}}(X,Y)$ be such that $\partial _T = \overline{CO} \{x\otimes y^*: y^*(T(x))= (x \otimes y^*)(T)=\Vert T\Vert ,~~x \in \partial _e X_1,~y^*\in \partial _e J^*_1\}$. An interesting question that arises from the previous section is that, when can one get a $x_0 \in \partial _e X_1$ such that $\Vert T\Vert =\Vert T(x_0)\Vert$ and $\partial _{T(x_0)}=\overline{CO}\{y^*\in \partial _e J^*_1: y^*(T(x_0))= \Vert T(x_0)\Vert \}$.

Since Y and hence J have the metric approximation property, we have that ${{\mathcal {K}}}(X,J)$ is a M-ideal in ${{\mathcal {K}}}(X,Y)$ (see [8] Proposition VI.3.1). Thus from the structure of extreme points of the dual unit ball of spaces of compact operators, we see that the description of $\partial _T$, as before, is in terms of the extreme points of the dual unit ball of the M-ideal ${{\mathcal {K}}}(X,J)$.

In what follows we will be using a minimax formula for compact operators. This author has recently proved ([9]), Theorem 1) the validity of this when J is a $L^1$-predual space (i.e., $J^*$ is isometric to a $L^1(\mu )$-space, for example, the spaces considered in Remark 8).

Theorem 13

Let X, Y, J be as above. Assume that X is a separable space, 0 is not a weak accumulation point of $\partial _e X_1$ and 0 is not a weak$^*$-accumulation point of $\partial _e Y^*_1$ and suppose the compact operator T satisfies the minimax formula, $d(T, {{\mathcal {K}}}(X,J))= sup\{d(T(x),J): x \in X_1\}$. If $\partial _T = \overline{CO}\{x\otimes y^*: (x \otimes y^*)(T) = \Vert T\Vert ,~x \in \partial _e X_1~,~y^*\in \partial _e J^*_1\}$, then there is a $x_0 \in \partial _e X_1$ such that, $\Vert T\Vert =\Vert T(x_0)\Vert$, $\partial _{T(x_0)} = \overline{CO}\{y^*\in \partial _e Y^*_1: y^*(T(x_0))=\Vert T(x_0)\Vert \}$.

Proof

Since X is reflexive and T is compact, let $x_0 \in \partial _e X_1$ be such that $\Vert T\Vert = \Vert T(x_0)\Vert$. Since ${{\mathcal {K}}}(X,J)$ is a M-ideal in ${{\mathcal {K}}}(X,Y)$, by our arguments given in the proof of Proposition 9, and our hypothesis implies, $d(T, {\mathcal {K}}(X,J))<\Vert T\Vert$. Now by the minimax formula, $d(T(x_0),J) < \Vert T\Vert =\Vert T(x_0)\Vert$. Now since J is a M-ideal, we get the description for $\partial _{T(x_0)}$. $\square$

For a compact Hausdorff space $\Omega$, let $C(\Omega ,Y)$ be the space of Y-valued continuous functions, equipped with the supremum norm. The minimax formula above was inspired by a classical result of Light and Cheney (see [14]), for a closed subspace $J \subset Y$ and $f \in C(\Omega ,Y)$,

$$\begin{aligned} d(f,C(\Omega ,J))= sup\{d(f(w),J): w \in \Omega \}. \end{aligned}$$

Analogous to the above theorem, if J is a M-ideal in Y, then $C(\Omega ,J)$ is a M-ideal in $C(\Omega ,Y)$ (see [8]). The following corollary, which assumes these conditions, is now easy to prove. For $\omega \in \Omega$, $\delta (\omega )$ denotes the Dirac measure at $\omega$.

Corollary 14

Suppose 0 is not a weak$^*$-accumulation point of $\partial _e Y^*_1$. Let $f \in C(\Omega ,Y)$ and suppose, $\partial _f = \overline{CO} \{\delta (\omega )\otimes y^*: y^*(f(\omega ))= \Vert f\Vert ~,~\omega \in \Omega ~,~y^*\in \partial _e J^*_1\}$. There exists a $\omega _0 \in \Omega$ such that $\Vert f\Vert =\Vert f(\omega _0)\Vert$, $\partial _{f(\omega _0)}= \overline{CO}\{y^*\in \partial _eJ^*_1: y^*(f(\omega _0))=\Vert f(\omega _0)\Vert \}$.

Let $WC(\Omega ,Y)$ denote the space of continuous functions when Y is considered with the weak topology, equipped with the supremum norm. We recall that for $f \in WC(\Omega ,Y)$, $\Vert f\Vert =Sup_{\omega \in \Omega ,y^*\in \partial _e Y^*_1} |y^*(f(\omega ))|$. Thus by an application of the separation theorem, $WC(\Omega ,Y)^*_1 = \overline{CO}\{\delta (\omega )\otimes y^*:\omega \in \Omega ~,~y^*\in \partial _e Y^*_1\}$. Now suppose 0 is not a weak$^*$-accumulation point of $\partial _e Y^*_1$, then as before we can conclude that for $f \in WC(\Omega ,Y)$, if $\Lambda \in \partial _e(\partial _f)$, then $\Lambda =\delta (\omega _0)\otimes y_0^*$ for some $\omega _0 \in \Omega$, $y_0^*\in \partial _e Y^*_1$. In particular, $\Vert f\Vert = \Vert f(\omega _0)\Vert$. We do not know how to prove a version of Corollary 14 in this set up.