An approximation problem in the space of bounded operators

TL;DR

This paper investigates approximation properties of bounded linear operators between Banach spaces, establishing conditions for global and local approximation equivalences, extremal attainment, and proximinality of subspaces.

Contribution

It provides new sufficient conditions for approximation equalities, extremal point attainment, and characterizes when certain operators are perpendicular to subspaces, extending previous results.

Findings

Conditions for the equality of global and local approximation distances.

Existence of extremal points where supremum is attained.

Characterization of perpendicularity involving bidual operators and subspace structures.

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 \cite{RAO21}, 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 $Y.$