Journal of Operator Theory

Volume 89, Issue 1, Winter 2023 pp. 49-74.

Compactness properties of multiplication and substitution operators

Authors: Laura Angeloni (1), Jurgen Appell (2), Tomas Dominguez (3), Simon Reinwand (4), Gianluca Vinti (5)
Author institution:(1) Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia, Perugia, 06123, Italy
(2) Mathematisches Institut, Universitat Wurzburg, Wurzburg, D-97074, Germany
(3) Facultad de Matematicas, Universidad de Sevilla, Sevilla, E-41080, Spain
(4) University of Wurzburg by TNG Technology Consulting GmbH, Beta-Str. 13a, D-85774 Unterfohring, Germany
(5) Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia, Perugia, 06123, Italy

Summary: In this paper we prove some conditions, mostly both necessary and sufficient, for the compactness of the multiplication operator $M_\mu(x)(t):= \mu(t)x(t)$ and the substitution operator $S_\varphi(x)(t):=x(\varphi(t))$ in the function spaces $C[0,1]$ and $BV[0,1]$. More general estimates for the essential norm and the measure of noncompactness of these operators are also obtained. A main emphasis is put on examples and counterexamples which illustrate the abstract results.

DOI: http://dx.doi.org/10.7900/jot.2021apr19.2333
Keywords: continuous functions, functions of bounded variation, multiplication operator, substitution operator, essential operator norm, measure of noncompactness

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