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1
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Operators of the q-oscillator

100%
Szafraniec F.
Banach Center Publications
|
2007
|
tom 78
|
nr 1
293-307
EN
We scrutinize the possibility of extending the result of [19] to the case of q-deformed oscillator for q real; for this we exploit the whole range of the deformation parameter as much as possible. We split the case into two depending on whether a solution of the commutation relation is bounded or not. Our leitmotif is subnormality. The deformation parameter q is reshaped and this is what makes our approach effective. The newly arrived parameter, the operator C, has two remarkable properties: it separates in the commutation relation the annihilation and creation operators from the deformation as well as it q-commutes with those two. This is why introducing the operator C may have far-reaching consequences.
2
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Subnormality versus restrictions

100%
Szafraniec F.
Banach Center Publications
|
2007
|
tom 75
|
nr 1
3
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Subnormality from bounded vectors

100%
Szafraniec F.
Banach Center Publications
|
2007
|
tom 75
|
nr 1
4
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Subnormality and cyclicity

100%
Szafraniec F.
Banach Center Publications
|
2005
|
tom 67
|
nr 1
349-356
EN
For an unbounded operator S the question whether its subnormality can be built up from that of every $S_f$, the restriction of S to a cyclic space generated by f in the domain of S, is analyzed. Though the question at large has been left open some partial results are presented and a possible way to prove it is suggested as well.
5
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Murphy's "Positive definite kernels and Hilbert C*-modules" reorganized

100%
Szafraniec F.
Banach Center Publications
|
2010
|
tom 89
|
nr 1
275-295
EN
The paper the title refers to is that in Proceedings of the Edinburgh Mathematical Society, 40 (1997), 367-374. Taking it as an excuse we intend to realize a twofold purpose: 1° to atomize that important result showing by the way connections which are out of favour, 2° to rectify a tiny piece of history. The objective 1° is going to be achieved by adopting means adequate to goals; it is of great gravity and this is just Mathematics. The other, 2°, comes from the author's internal need of showing how ethical values in Mathematics are getting depreciated. The latter has nothing to do with the previous issue; the coincidence is totally accidental.
6
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Notes on q-deformed operators

64%
Ôta S., Szafraniec F.
Studia Mathematica
|
2004
|
tom 165
|
nr 3
295-301
EN
The paper concerns operators of deformed structure like q-normal and q-hyponormal operators with the deformation parameter q being a positive number different from 1. In particular, an example of a q-hyponormal operator with empty spectrum is given, and q-hyponormality is characterized in terms of some operator inequalities.
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