On \(k\)-smoothness of operators between Banach spaces (Q6601143)

This article develops the study of \(k\)-smoothness of operators between Banach spaces. This study brings new insights to the geometry of the unit sphere of operator spaces. The authors review relevant definitions in the Introduction and set notation to be followed throughout. We list the following notation and definitions from their introduction: \(X\) and \(Y\) denote Banach spaces over the field \(K\) (that represents either the field of complex or the field of real numbers). \(L(X,Y)\) (resp. \(K(X,Y)\)) denotes the spaces of all bounded (resp. compact) operators from \(X\) to \(Y\). For \(T\in L(X,Y)\),\N\[\NM_T=\{ x\in X: \|x\|=1, \ \|x\|=\|Tx\|\}.\N\]\NThe authors also recall the definitions of extreme point of a convex subset \(A\) of \(X\), \(\mathrm{Ext} (A)\), smoothness, \(k\)-smoothness, supporting functional of an \(x \in X\), the set of all supporting functionals is denoted by \(J(x)\), $M$-ideals, and Birkhoff-James orthogonality of two elements in \(X\): \(x,y \in X\) are B-orthogonal (\(x\perp_B y\)) iff for all \(\lambda \in K\), \(\|x+\lambda y\|\geq \|x\|.\)\N\NLet \(X,Y\) be Banach spaces and let \(T \in L(X,Y)\) of norm~1. Let \(R\) be a subset of the unit sphere $S_X$ of \(X\) such that \(R\cap \mathrm{Ext}(B_X)\neq \emptyset\). Let \(\{v_1,v_2,\dots ,v_n\} \) be a basis of $\operatorname{span}R\). Suppose that\N\[\NW = \mathrm{span} \{ y^* \in J(Tv): v \in R\cap \mathrm{Ext}(B_X)\}\N\]\Nis a finite-dimensional subspace of \(Y^{\star} \) and let \(\{y_1^*,y_2^*, \ldots, y_p^*\}\) be a basis for \(W\). Then the index of smoothness of \(T\) with respect to \(R\) is denoted by \(i_R(T)\), and it is defined to be the dimension of the span of\N\[\NZ= \Bigl\{(\alpha_i, \beta_j)_{1\leq i\leq n, 1\leq j\leq p}\in K^{pn}: \sum_{i=1}^n \alpha_i v_i \in R\cap \mathrm{Ext}(B_X),\ \sum_{j=1}^p \beta_jy_j^* \in \mathrm{Ext}\Bigl(J\Bigl(\sum_{i=1}^n \alpha_i Tv_i \Bigr)\Bigr) \Bigr\}.\N\]\NThe authors also mention that the index of smoothness of $T$ with respect to $R$ depends neither on the choice of basis of $\operatorname{span} R$ nor on the basis of $W$. One of the first major results in the paper is described next.\N\N{Theorem (2.2).} Let \(X\) be a reflexive strictly convex Banach space and let \(Y\) be a smooth Banach space. Suppose that \(T\in L(X,Y )\) is of norm 1, and \(K(X, Y )\) is an $M$-ideal of \(L(X, Y ) \) with \(\mathrm{dist}(T,K(X,Y)) < 1.\) Then \(g \in J(T)\) is an element of \(\mathrm{Ext}(J(T))\) if and only if there exists a fixed element \(x_0 \in M_T\) such that \(T (x_0) \perp_B Ax_0,\) for every \( A \in \ker g\).\N\NThe authors also include an interesting theorem due to Bhatia-Šemrl involving the same notions and a new result strengthening the theorem under some conditions:\N\N{Theorem (2.5).} Let \(X\) be a reflexive strictly convex Banach space and let \(Y\) be a smooth Banach space. Suppose that \(T\in L(X,Y )\) is of norm 1, and \(K(X, Y )\) is an $M$-ideal of \(L(X, Y ) \) with \(\mathrm{dist}(T,K(X,Y)) < 1.\) Then\N\N(i) for a subspace \(W\) of \( L(X,Y),\) there exists a fixed \(x_0 \in M_T\) such that \(Tx_0 \perp_B A x_0\) , for any \(A\in W\) if and only if \(W\subset \ker g, \) for some \( g\in \mathrm{Ext}(J(T))\); \N\N(ii) a subspace \(W \) of \(L(X,Y)\) satisfies \(Tx \perp_B Ax,\) for every \(x \in M_T\) and for every \( A\in W\) if and only if \( W \subset \bigcap_{g \in \mathrm{Ext}(J(T))}\ker g.\)\N\NA complete characterization of $k$-smoothness for Hilbert space operators is due to \textit{A.~Mal} et al. [Linear Multilinear Algebra 70, No.~18, 3477--3489 (2022; Zbl 1511.46012)]. The authors say that the notions developed in the article extend the result for operators between Banach spaces as follows:\N\N{Theorem (2.16).} Let \( X \) and \( Y\) be strictly convex, smooth, reflexive Banach spaces. Suppose that \(T \in L(X,Y )\) is of norm 1, and \(K(X, Y )\) is an $M$-ideal of \(L(X, Y ) \) with \(\mathrm{dist}(T,K(X,Y)) < 1.\) Let \(M_T \) be the unit sphere of an \(n\)-dimensional subspace \( X_0\) of \( X\). Suppose that \(T(X_0)\) is a coproximinal subspace of \(Y\) and \(T(X_0)\) has a strong Auerbach basis. Then\N\N(i) $T$ is at least $\binom{n+1}2$ \(\left( \begin{array}{c} n+1 \\\N2 \end{array} \right) \)-smooth;\N\N(ii) \(T\) is \(n^2\)-smooth when \(X,Y\) are over the complex field.\N\NSeveral consequences follow from Theorem~2.16, like for example a characterization for the real Hilbert spaces among the \(\ell^n_p (\mathbb{R}) \) spaces, and also a characterization of the $k$-smoothness of an element in a finite-dimensional polyhedral Banach space. The paper also includes several illustrative examples.