Extension of $p$-compact operators in Banach spaces
Sainik Karak, Tanmoy Paul
TL;DR
The paper addresses when $p$-compact and weakly $p$-compact operators can be extended from a Banach space $X$ to larger domains $Z$ while preserving their compactness properties and norms, particularly under the condition that $X^{**}$ is a $P_\lambda$-space. It develops extension results with explicit norm bounds $\kappa_p(\widetilde{T}) \le \lambda\kappa_p(T)$ and $\kappa_p^d(\widetilde{T}) \le \lambda\kappa_p^d(T)$ (and analogous $\omega_p$-bounds) and provides dual-extension constructions via factorization, along with a norm-preserving extension in the compact-case for finite codimension. The work connects these extensions to the MAP property and to $L_1$-predual structures, and introduces a universal $P_1$-space framework for dual extensions. Together, these results clarify when operator-ideal extensions retain $p$-compactness and quantify how the extension process impacts the associated norms, contributing to the theory of Banach space extensions and the structure of $L_1$-preduals.
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 $P_λ$ 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.