Model spaces invariant under composition operators. Issue 1 (25th March 2023)

• Record Type: Journal Article
• Authors: Muthukumar, P.; Sarkar, Jaydeb
• Is Part Of: Canadian mathematical bulletin =. Volume 66:Issue 1(2023)
• Page Start: 204
• Page End: 217
• Publication Date: 2023-03-25
• Subjects: 47B33 -- 30J05 -- 47B91 -- 30H10 -- 30D55 -- 46E15 -- 47A15 -- 46E22 -- 30J10; Composition operators -- inner functions -- model spaces -- Hardy space -- fractional linear transformations -- Blaschke products; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439522000236
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: Given a holomorphic self-map $\varphi $ of $\mathbb {D}$ (the open unit disc in $\mathbb {C}$ ), the composition operator $C_{\varphi } f = f \circ \varphi $, $f \in H^2(\mathbb {\mathbb {D}})$, defines a bounded linear operator on the Hardy space $H^2(\mathbb {\mathbb {D}})$ . The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb {\mathbb {D}})$, which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.