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Unbounded weighted shifts and subnormality. (English) Zbl 0675.47011

The operator \(S=(1/\sqrt{2})(x-d/dx)\) defined on the linear span of the Hermite functions is an example of an unbounded subnormal weighted shift [cf. the authors, J. Oper. Theory 14, 31–55 (1985; Zbl 0613.47022)]. It is shown that certain products of unbounded subnormal weighted shifts are again subnormal, for example the products \(R^{*pt}S^{pt}T\), \(TR^{pt}S^{*pt}\), \(TR^{*pt}S^{pt}\), \(R^{pt}S^{*pt}T\) for integers \(p\geq 0\), \(t\geq 1\) if \(R,S\) are usual weighted shifts and \(T\) is a weighted \(t\)-shift (i.e. it shifts \(t\)-times)

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Citations:

Zbl 0613.47022
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Full Text: DOI

References:

[1] Ch. Berg, J.P.R. Christensen, P. Ressel,Harmonic analysis on semigroups, Springer, Berlin, 1984.
[2] A.L. Shields, Weighted shifts and analytic function theory,Mathematical Surveys, vol. 13, American Mathematical Society, Providence, RI, 1974. · Zbl 0303.47021
[3] J. Stochel and F.H. Szafraniec, Bounded vectors and formally normal operators, in ?Dilation Theory, Toeplitz Operators and Other Topics?, eds. C. Apostol, C.M. Pearcy, B.Sz.-Nagy and D. Voiculescu,Operator Theory: Advances and Applications, vol. 11, pp. 363-370, Birkhäuser, Basel, 1983. · Zbl 0544.47020
[4] ?, On normal extensions of unbounded operators. I,J. Operator Theory, 14 (1985), 31-55. · Zbl 0613.47022
[5] J. Stochel and F.H. Szafraniec, On normal extensions of unbounded operators. II, Institute of Math., Polish Academy of Sciences, Preprint No. 349, Nov. 1985; to appear inActa Sci. Math. (Szeged). · Zbl 0613.47022
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