Journal of Operator Theory

Volume 54, Issue 1, Summer 2005 pp. 169-187.

Invariant subspaces for commuting pairs with normal boundary dilation and dominating Taylor spectrum

Authors: Michael Didas
Author institution: Universit\''at des Saarlandes, Fachrichtung 6.1 -- Mathematik, Postfach 15 11 50, D-66041 Saarbr\''ucken, Germany

Summary: Let $T \in L(H)^n$ be a commuting tuple of continuous linear operators on a separable complex Hilbert space. In this article we show that interior points of the Fredholm spectrum of $T$ can be made accessible to the Scott Brown technique by establishing factorizations of the corresponding point evaluations via the holomorphic functional calculus. This allows us to improve a series of known results in the context of the invariant-subspace and the reflexivity problem. In particular we deduce that each commuting pair $T=(T_1, T_2) \in L(H)^2$ possessing a $\partial D$-unitary dilation and dominating Taylor spectrum in a strictly pseudoconvex open subset $D \Subset \C^2$ has a non-trivial invariant subspace.

Keywords: $H^\infty$-functional calculus, invariant subspaces, dominating Taylor spectrum, strictly pseudoconvex sets.

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