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On Extending Commutative Semigroups of Isometries

On Extending Commutative Semigroups of Isometries ON EXTENDING COMMUTATIVE SEMIGROUPS OF ISOMETRIES R. G. DOUGLAS Let V be an isometric operator defined on the Hilbert space «#, that is, ||Kx|| = Hxll for x in «5f. From a result due t o von Neumann [5] and Wold [7], it follows that there is a unitary operator W defined on a Hilbert space J f containing #? that extends V. An analogous result was obtained by Cooper [2] for a con- tinuous one parameter semi-group of isometries. Independently, Ito [4] and Brehmer [1] showed that every commutative semigroup of isometries on Hilbert space can be extended to a corresponding commutative semigroup of unitary operators on a larger Hilbert space. It is the purpose of this note to give a more direct and natural proof of this latter result which is valid for Banach spaces and to prove certain ancillary results concerning the commutant of the semigroup of isometries. The proof is based on the construction of the direct limit of Banach spaces. A precise statement of the result will be given after this construction has been carried out. Let £ be a commutative semigroup and 3C be a Banach space. An isometric representation of £ on http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On Extending Commutative Semigroups of Isometries

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References (4)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/1.2.157
Publisher site
See Article on Publisher Site

Abstract

ON EXTENDING COMMUTATIVE SEMIGROUPS OF ISOMETRIES R. G. DOUGLAS Let V be an isometric operator defined on the Hilbert space «#, that is, ||Kx|| = Hxll for x in «5f. From a result due t o von Neumann [5] and Wold [7], it follows that there is a unitary operator W defined on a Hilbert space J f containing #? that extends V. An analogous result was obtained by Cooper [2] for a con- tinuous one parameter semi-group of isometries. Independently, Ito [4] and Brehmer [1] showed that every commutative semigroup of isometries on Hilbert space can be extended to a corresponding commutative semigroup of unitary operators on a larger Hilbert space. It is the purpose of this note to give a more direct and natural proof of this latter result which is valid for Banach spaces and to prove certain ancillary results concerning the commutant of the semigroup of isometries. The proof is based on the construction of the direct limit of Banach spaces. A precise statement of the result will be given after this construction has been carried out. Let £ be a commutative semigroup and 3C be a Banach space. An isometric representation of £ on

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1969

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