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Subnormality and generalized commutation relations of families of operators†

Published online by Cambridge University Press:  18 May 2009

Jerzy Bartłomiej Stochel
Jerzy Bartłomiej Stochel
Affiliation:
Institute of Mining and MetallurgyAl Mickiewicza 3030–059 Krakow, Poland
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1. Every family of subnormal operators in a Hilbert space fulfils the Halmos-Bram condition on a suitable dense subset of its domain [2], [3]. In [2] and [4] it is shown that the generalized commutation relation implies the Halmos-Bram condition for one operator. In this paper it is proved that the generalized commutation relation implies the Halmos-Bram condition for infinite families of operators (in a special case Jorgensen proved it in a different way for finite families of operators, see [2]) and as an example of the application of this property it is shown that every family of generalized creation operators in the Bargmann space of an infinite order, indexed by mutually orthogonal vectors from I2 is subnormal. See [1] for the definitions.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 2 , May 1990 , pp. 231 - 238
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

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3.Stochel, J. and Szafraniec, F. H., On normal extensions of unbounded operators I J. Operator Theory 14 (1985), 31–55.Google Scholar
4.Stochel, J. B., Subnormality and generalized commutation relations, Glasgow Math. J. 30 (1988), 259–262.CrossRefGoogle Scholar
5.Stochel, J. B., Subnormality of generalized creation operators in Bargmann's Hilbert space of an infinite order, in preparation.Google Scholar