Details
| Original language | English |
|---|---|
| Article number | 1630001 |
| Pages (from-to) | 1630001 |
| Number of pages | 1 |
| Journal | Rev. Math. Phys. |
| Volume | 28 |
| Issue number | 3 |
| Publication status | Published - 1 Apr 2016 |
Abstract
In this paper, we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them, in particular, with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Fréchet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show, in particular, that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally, we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators.
Keywords
- Quantum harmonic analysis, Schwartz functions, canonical commutation relations
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematics(all)
- Mathematical Physics
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In: Rev. Math. Phys., Vol. 28, No. 3, 1630001, 01.04.2016, p. 1630001.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Schwartz operators
AU - Keyl, Michael
AU - Kiukas, Jukka
AU - Werner, Reinhard F.
N1 - Publisher Copyright: © 2016 World Scientific Publishing Company. Copyright: Copyright 2016 Elsevier B.V., All rights reserved.
PY - 2016/4/1
Y1 - 2016/4/1
N2 - In this paper, we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them, in particular, with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Fréchet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show, in particular, that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally, we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators.
AB - In this paper, we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them, in particular, with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Fréchet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show, in particular, that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally, we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators.
KW - Quantum harmonic analysis
KW - Schwartz functions
KW - canonical commutation relations
UR - http://www.scopus.com/inward/record.url?scp=84966648769&partnerID=8YFLogxK
U2 - 10.1142/S0129055X16300016
DO - 10.1142/S0129055X16300016
M3 - Article
VL - 28
SP - 1630001
JO - Rev. Math. Phys.
JF - Rev. Math. Phys.
SN - 1793-6659
IS - 3
M1 - 1630001
ER -