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

  • W. E. Longstaff (a1)

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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.

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References

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1. Birkhoff, G., Lattice theory (Amer. Math. Soc. Colloquium Publications, Vol. XXV 1948).
2. Birkhoff, G., On the structure of abstract algebras, Proc. Camb. Phil. Soc. 31 (1935), 433454.
3. Brickman, L. and Fillmore, P. A., The invariant subspace lattice of a linear transformation, Can. J. Math. 19 (1967), 810822.
4. Fillmore, P. A., Herrero, Domingo A. and Longstaff, W. E., The hyper invariant subspace lattice of a linear transformation, preprint.
5. Hoffman, K. and Kunze, R., Linear algebra (Second edition, Prentice-Hall, N.J., 1971).
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A Lattice-Theoretic Description of the Lattice of Hyperinvariant Subspaces of a Linear Transformation

  • W. E. Longstaff (a1)

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