Abstract
In this essay we try to explain what may happen if one wants a two-sided weighted shift in a Hilbert space to be Krein normal. As in principle this is not the case it turns out to be so provided the definition of a Krein space is extended in away which is both natural and provocative; the extended notion may be looked at as a sort of ‘complexification’ of the classical one. The aforesaid desire has come out of an attempt at classifying the odd solution of the commutation relation of the q-oscillator, which appears in the most innocent case of 0 < q < 1. A more prosaic motivation is as follows: the two-sided shift is unitary, hence normal; what kind of normality may be attributed to a two-sided weighed shift?
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Szafraniec, F.H. (2008). Two-sided Weighted Shifts Are ‘Almost Krein’ Normal. In: Behrndt, J., Förster, KH., Langer, H., Trunk, C. (eds) Spectral Theory in Inner Product Spaces and Applications. Operator Theory: Advances and Applications, vol 188. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8911-6_13
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DOI: https://doi.org/10.1007/978-3-7643-8911-6_13
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