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Journal of Operator Theory

Volume 73, Issue 1, Winter 2015  pp. 243-264.

Invariant subspaces of composition operators

Authors:  Valentin Matache
Author institution:Department of Mathematics, University of Nebraska, Omaha NE, 68182, U.S.A.

Summary: The invariant subspace lattices of composition operators acting on $H^2$, the Hilbert-Hardy space over the unit disc, are characterized in select cases. The lattice of all spaces left invariant by both a composition operator and the unilateral shift $M_z$ (the multiplication operator induced by the coordinate function), is shown to be nontrivial and is completely described in particular cases. Given an analytic selfmap $\varphi $ of the unit disc, we prove that $\varphi $ has an angular derivative at some point on the unit circle if and only if $C_\varphi $, the composition operator induced by $\varphi $, maps certain subspaces in the invariant subspace lattice of $M_z$ into other such spaces. A similar characterization of the existence of angular derivatives of $\varphi $, this time in terms of $A_\varphi $, the Aleksandrov operator induced by $\varphi $, is obtained.

Keywords: composition operator, invariant subspaces

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