Extension of $p$-compact operators in Banach spaces

TL;DR

This paper investigates the extension properties of p-compact operators in Banach spaces, especially when spaces are P_lambda spaces, and explores conditions for almost norm-preserving extensions and L_1-preduality.

Contribution

It provides new criteria for extending p-compact operators and their adjoints in Banach spaces, particularly in P_lambda spaces, and characterizes L_1-predual spaces.

Findings

Operators can often be extended to larger domains with appropriate codomain extensions.

Extensions can be almost norm-preserving under certain conditions.

Characterization of L_1-predual Banach spaces.

Abstract

We analyze various consequences in relation to the extension of operators $T:X\to Y$ that are $p$-compact, as well as the extension of operators $T:X\to Y$ whose adjoints $T^*:Y^*\to X^*$ are $p$-compact. In most cases, we discuss these extension properties when the underlying spaces, either domain or codomain, are $P_\lambda$ spaces. We also answer if these extensions are almost norm-preserving in such circumstances where the extension $\widetilde{T}$ of a $T$ exists. It is observed that an operator can often be extended to a larger domain when the codomain is appropriately extended as well. Specific assumptions might enable us to obtain an extension of an operator that maintains the same range. Necessary and sufficient conditions are derived for a Banach space to be $L_1$-predual.