Maximal invariant subspaces for a class of operators

Author： Guo Kunyu  

Publisher： Springer Publishing Company

ISSN： 0004-2080

Source： Arkiv för Matematik, Vol.48, Iss.2, 2010-10, pp. : 323-333

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Abstract

In this note, we characterize maximal invariant subspaces for a class of operators. Let T</i> be a Fredholm operator and $1-TT^{*}inmathcal{S}_{p}$</EquationSource> for some p</i>≥1. It is shown that if M</i> is an invariant subspace for T</i> such that dim M</i> ⊖</i> TM</i><∞, then every maximal invariant subspace of M</i> is of codimension 1 in M</i>. As an immediate consequence, we obtain that if M</i> is a shift invariant subspace of the Bergman space and dim M</i> ⊖</i> zM</i><∞, then every maximal invariant subspace of M</i> is of codimension 1 in M</i>. We also apply the result to translation operators and their invariant subspaces.

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