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INVARIANT SUBSPACES FOR PAIRS OF PROJECTIONS

Published online by Cambridge University Press:  01 April 1998

GRAHAM R. ALLAN  and
JAROSLAV ZEMÁNEK
GRAHAM R. ALLAN
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB. E-mail: [email protected]
JAROSLAV ZEMÁNEK
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 137, 00-950 Warszawa, Poland. E-mail: [email protected]
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Abstract

A simple geometrical argument shows that every pair of projections on a finite-dimensional complex vector space has a common invariant subspace of dimension 1 or 2. The idea extends to certain pairs of projections on an infinite-dimensional Hilbert space H. In particular every projection on H has a reducing subspace, although a finite-dimensional one need not exist. In a final section, the results are extended to the existence of hyperinvariant subspaces for pairs of projections.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 57 , Issue 2 , April 1998 , pp. 449 - 468
Copyright
The London Mathematical Society 1998

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