On reducibility of weighted composition operators

Document Type: Research Paper

Authors

1 Department of Mathematics, University of Maragheh, Maragheh, Iran.

2 faculty of mathematical sciences, university of tabriz, p. o. box: 5166615648, tabriz,

Abstract

In this paper, we study two types of the reducing subspaces for the weighted composition operator $W: f\rightarrow u\cdot f\circ \varphi$ on $L^2(\Sigma)$. A necessary and sufficient condition is given for $W$ to possess the reducing subspaces of the form $L^2(\Sigma_B)$ where $B\in \Sigma_{sigma(u)}$. Moreover, we pose some necessary and some sufficient conditions under which the subspaces of the form $L^2(\mathcal{A})$ reduce $W$. All of these are basically discussed by using the conditional expectation properties. To explain the results some examples are then presented.

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