A Survey of Some Results on Invariant Subspaces in Operator Theory

This concise monograph explores how core ideas in Hardy space function theory and operator theory continue to be useful and informative in new settings, leading to new insights for noncommutative multivariable operator theory. Beginning with a review of the confluence of system theory ideas and reproducing kernel techniques, the book then covers representations of backward-shift-invariant subspaces in the Hardy space as ranges of observability operators, and representations for forward-shift-invariant subspaces via a Beurling–Lax representer equal to the transfer function of the linear system. This pair of backward-shift-invariant and forward-shift-invariant subspace form a generalized orthogonal decomposition of the ambient Hardy space. All this leads to the de Branges–Rovnyak model theory and characteristic operator function for a Hilbert space contraction operator. The chapters that follow generalize the system theory and reproducing kernel techniques to enable an extension of the ideas above to weighted Bergman space multivariable settings.

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Invariant Complements to Closed Invariant Subspaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-015-1 ◽

1979 ◽

Vol 31 (1) ◽

pp. 139-147 ◽

Cited By ~ 1

Author(s):

M. P. Thomas

Keyword(s):

Invariant Subspace ◽

Operator Theory ◽

Point Spectrum ◽

Volterra Operator ◽

Invariant Subspaces ◽

Dense Range ◽

Mild Conditions ◽

Closed Invariant Subspace ◽

Quasinilpotent Operator

The question under what conditions a closed invariant subspace possesses a closed invariant complement is of major importance in operator theory. In general it remains unanswered. In this paper we drop the requirement that the invariant complement be closed. We show in section 1 that the question is answerable under fairly mild conditions for a quasinilpotent operator (Theorem 1.5). These conditions will cover the case of a quasinilpotent operator with dense range and no point spectrum. In section 2 we discuss the consequences for the Volterra operator V. Since V is unicellular, its proper closed invariant subspaces do not possess closed invariant complements. However, they are all algebraically complemented (Proposition 2.1).

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