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Dissipative Operators and Approximation of Inverses

Dissipative Operators and Approximation of Inverses DISSIPATIVE OPERATORS AND APPROXIMATION OF INVERSES J. A. ERDOS A result in [4] (p. 16) attributed to Brodskii states that if T is a compact injective quasinilpotent dissipative operator then every maximal nest of subspaces invariant under T is continuous. It is proved below (Theorem 3) that this result remains true without the condition that T is compact. The proof is simpler than that given in [4] for the compact case. (Note that the result does not imply the existence of any invariant subspace so the statement may be vacuous for some operators.) The ideas are then applied to the following question considered by Feintuch [1, 2, 3]: under what con- ditions is the inverse of an invertible operator A a weak limit of polynomials in A ? In particular a strengthening of the result of [3] is given. Standard terminology and notation will be used (see, for example [5]). The terms Hilbert space, subspace and operator will be used to mean complex Hilbert space, closed subspace and continuous linear operator on a Hilbert space respectively. The operator T = X + iY (where X and Y are the usual real and imaginary parts of T ) is said http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Dissipative Operators and Approximation of Inverses

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/11.2.142
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Abstract

DISSIPATIVE OPERATORS AND APPROXIMATION OF INVERSES J. A. ERDOS A result in [4] (p. 16) attributed to Brodskii states that if T is a compact injective quasinilpotent dissipative operator then every maximal nest of subspaces invariant under T is continuous. It is proved below (Theorem 3) that this result remains true without the condition that T is compact. The proof is simpler than that given in [4] for the compact case. (Note that the result does not imply the existence of any invariant subspace so the statement may be vacuous for some operators.) The ideas are then applied to the following question considered by Feintuch [1, 2, 3]: under what con- ditions is the inverse of an invertible operator A a weak limit of polynomials in A ? In particular a strengthening of the result of [3] is given. Standard terminology and notation will be used (see, for example [5]). The terms Hilbert space, subspace and operator will be used to mean complex Hilbert space, closed subspace and continuous linear operator on a Hilbert space respectively. The operator T = X + iY (where X and Y are the usual real and imaginary parts of T ) is said

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Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1979

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