Volterra and Composition Inner Derivations on the Fock–Sobolev Spaces

Abstract

On the Fock–Sobolev spaces, we study the range of Volterra inner derivations and composition inner derivations. The Volterra inner derivation ranges in the ideal of compact operators if and only if the induced function g is a linear polynomial. The composition inner derivation ranges in the ideal of compact operators if and only if the induced function φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} is either identity or a contractive linear self-mapping of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}$$\\end{document}. Moreover, we describe the compact intertwining relations for composition operators and Volterra operators between different Fock–Sobolev spaces. In this paper, our results are complement and in a sense extend some aspects of Calkin’s result (Ann Math 42:839–873, 1941) to the algebras of bounded linear operators on Fock–Sobolev spaces.