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Remarks on Invariant Subspace Lattices

Published online by Cambridge University Press:  20 November 2018

Peter Rosenthal
Peter Rosenthal*
Affiliation:
University of Toronto
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If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 5 , October 1969 , pp. 639 - 643
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bernstein, A.R. and Robinson, A., Solution to an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 (1966) 421–431.Google Scholar
2. Brickman, L. and Fillmore, P. A., The invariant subspace lattice of a linear transformation. Canad. J. Math. 19 (1967) 810–822.Google Scholar
3. Halmos, P.R., Invariant subspaces. Proc. 1968 Oberwolfoch Conference on Approximation Theory (to appear).Google Scholar
4. Hoffman, K., Banach spaces of analytic functions. (Prentice-Hall, Englewood Cliffs, 1962).Google Scholar
5. Rosenthal, Peter, Examples of invariant subspace lattices. Duke Math. J. (to appear).Google Scholar