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A Lattice-Theoretic Description of the Lattice of Hyperinvariant Subspaces of a Linear Transformation

Published online by Cambridge University Press:  20 November 2018

W. E. Longstaff
W. E. Longstaff*
Affiliation:
University of Western Australia, Nedlands, Western Australia
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If A is a (linear) transformation acting on a (finitedimensional, non-zero, complex) Hilbert space H the family of (linear) subspaces of H which are invariant under A is denoted by Lat A. The family of subspaces of H which are invariant under every transformation commuting with A is denoted by Hyperlat A. Since A commutes with itself we have Hyperlat A ⊆ Lat A. Set-theoretic inclusion is an obvious partial order on both these families of subspaces. With this partial order each is a complete lattice; joins being (linear) spans and meets being set-theoretic intersections. Also, each has H as greatest element and the zero subspace (0) as least element. With this lattice structure being understood, Lat A (respectively Hyperlat A) is called the lattice of invariant (respectively, hyper invariant) subspaces of A.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 5 , 01 October 1976 , pp. 1062 - 1066
Copyright
Copyright © Canadian Mathematical Society 1976

References

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