Essential norms of pointwise multipliers in the non-algebraic setting

TL;DR

This paper computes the essential and weak essential norms of multiplication operators between different K{" o}the spaces over the same measure space, with applications to various Banach spaces including Hardy, Musielak--Orlicz, Lorentz, and Marcinkiewicz spaces.

Contribution

It provides explicit formulas for essential norms of multiplication operators in a non-algebraic setting and applies these results to a broad class of Banach sequence and function spaces.

Findings

Explicit formulas for essential norms of multiplication operators.

Applications to Hardy, Musielak--Orlicz, Lorentz, and Marcinkiewicz spaces.

Illustrative examples demonstrating the theoretical results.

Abstract

Motivated by some recent results, but also referring to recognized classics, we compute the essential norm and the weak essential norm of multiplication operators acting between two distinct K{\" o}the spaces both defined over the same $\sigma$-finite measure space. A by-product of the technology we have developed here are some applications to Banach sequence spaces related to decreasing functions and to Banach spaces of analytic functions on the unit disc, in particular, Hardy spaces. We will close our work with some specific examples illustrating the previously obtained results including Musielak--Orlicz sequence spaces (in particular, Nakano sequence spaces) as well as Lorentz and Marcinkiewicz sequence spaces.