Opuscula Math. 26, no. 1 (2006), 151-160

Opuscula Mathematica

# A remark on generalized commutation relation and subnormality

Jerzy Bartłomiej Stochel

Abstract. Tillmann [Tillmann H. G., Zur Eindeutigkeit der Losungen der quanten mechanischen vertauschungrelationen, Acta Sci. Math. (Szeged) 24 (1963), 258-270] proved that every operator $$A$$ which fulfils the canonical commutation relation $$A^{*}A - AA^{*} = Id$$ is an orthogonal sum of canonical creation operators. We extend this result for operators which fulfil generalized commutation relation $A^{*}A - AA^{*}= E^2,\text{ where }EA = AE.$ In addition, some inequalities which are helpful in describing analytic vectors of operators $$A^{*}A$$, where $$A$$ fulfils the generalized commutation relation, are established.

Keywords: Hilbert space, generalized commutation relation, creation operator, analytic vectors.

Mathematics Subject Classification: 47B20, 47B37, 47B47.

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• Jerzy Bartłomiej Stochel
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
• Received: 2005-02-15.

Cite this article as:
Jerzy Bartłomiej Stochel, A remark on generalized commutation relation and subnormality, Opuscula Math. 26, no. 1 (2006), 151-160

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