TY - JOUR

T1 - Schwartz operators

AU - Keyl, Michael

AU - Kiukas, Jukka

AU - Werner, Reinhard. F.

N1 - Publisher Copyright:
© 2016 World Scientific Publishing Company.

PY - 2016/5/3

Y1 - 2016/5/3

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 - canonical commutation relations

KW - Schwartz functions

KW - Quantum harmonic analysis

UR - http://hdl.handle.net/2160/44092

UR - http://www.scopus.com/inward/record.url?scp=84966648769&partnerID=8YFLogxK

U2 - 10.1142/S0129055X16300016

DO - 10.1142/S0129055X16300016

M3 - Review Article

SN - 0129-055X

VL - 28

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

IS - 3

M1 - 1630001

ER -