Schwartz operators

Keyl, Michael; Kiukas, Jukka; Werner, R. F.

doi:10.1142/s0129055x16300016

Cited by 39 publications

(64 citation statements)

References 77 publications

(126 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We will now introduce an analogue of the Fourier transform for a trace class operator S. The Fourier-Wigner transform F W S of S is the function given by F W S(z) = e −πix·ω tr(π(−z)S) for z ∈ R 2d . In the terminology of Werner [25,26,34] this is the Fourier-Weyl transform, but we follow Folland [18] and call it the Fourier-Wigner transform.…”

Section: A Fourier Transform For Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

Convolutions for localization operators

Luef

Skrettingland

2018

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

Quantum harmonic analysis on phase space is shown to be linked with localization operators. The convolution between operators and the convolution between a function and an operator provide a conceptual framework for the theory of localization operators which is complemented by an appropriate Fourier transform, the Fourier-Wigner transform. We link the Hausdorff-Young inequality for the Fourier-Wigner transform with Lieb's inequality for ambiguity functions. Noncommutative Tauberian theorems due to Werner allow us to extend results of Bayer and Gröchenig on localization operators. Furthermore we show that the Arveson spectrum and the theory of Banach modules provide the abstract setting of quantum harmonic analysis.1991 Mathematics Subject Classification. 47G30; 35S05; 46E35; 47B10.

show abstract

Section: A Fourier Transform For Operatorsmentioning

confidence: 99%

“…Using these concepts we formulate and prove a version of Wiener's Tauberian theorem for operators due to Werner. These variants of Tauberian theorems have shown to be of relevance in quantum mechanics and quantum information theory [25,26].…”

Section: Introductionmentioning

confidence: 96%

Convolutions for localization operators

Luef

Skrettingland

2018

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

show abstract

“…We let S be the set of pseudodifferential operators with Weyl symbol in the Schwartz class S(R 2d ), and we let S ′ be the set of pseudodifferential operators with Weyl symbol in the tempered distributions S ′ (R 2d ). These sets of operators were studied in detail by Keyl et al in [43]. They show that S may be equipped with a topology making it a Frechet space, and that S ′ is the topological dual space of S in this topology.…”

Section: Localization Operators and Spectrograms As Convolutionsmentioning

confidence: 99%

“…Hence one may define convolutions and Fourier transforms on S ′ using duality. We summarize the main results in the following proposition, and refer to section 5 of [43] for proofs.…”

Section: Localization Operators and Spectrograms As Convolutionsmentioning

confidence: 99%

Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

Luef

Skrettingland

2019

J Fourier Anal Appl

View full text Add to dashboard Cite

We study mixed-state localization operators from the perspective of Werner's operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore we provide a description of the Cohen class in terms of Werner's convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen's class.where π(z)ψ(t) = e 2πiωt ψ(t−x). In [55] we showed that this yields a natural class of Banach modules. There are two types of convolutions in this noncommutative setting: (i) The convolution between a function f ∈ L 1 (R 2d ) and a trace class operator S:(ii) the convolution between two trace class operators S and T is defined by1991 Mathematics Subject Classification. 47G30; 35S05; 46E35; 47B10. Key words and phrases. localization operators, Cohen class, uncertainty principle, phase retrieval, positive operator valued measures.F W S(z) = e −πix·ω tr(π(−z)S) for z ∈ R 2d . Note that the Fourier-Wigner transform and the spreading function differ only by a phase factor. The Fourier-Wigner transform has many properties analogous to those of the Fourier transform of functions [55,66]. In the case of rank-one operators these concepts of quantum harmonic analysis turn into wellknown objects from time-frequency analysis. Suppose ϕ 2 ⊗ ϕ 1 is the rank-one operator for ϕ 1 , ϕ 2 ∈ L 2 (R d ). Then we havewhich is a localization operator (or STFT-filter or 46]) and is denoted by A ϕ1,ϕ2f , and f is called the mask of the STFT-filter. Similarly, the convolution of two rank-one operators becomeswhereξ(x) = ξ(−x), which reduces for η = ψ and ψ = φ to the spectrogram [42]. The Fourier-Wigner transform of a rank-one operator is the ambiguity function. There is also a Hausdorff-Young inequality associated to the Fourier-Wigner transform [55,66], that in the rank-one case is the non-sharp Lieb's inequality for ambiguity functions [51]. Let us return to the objectives of this paper. Since localization operators are convolutions of a function and a rank-one operator, a natural extension of localization operators are operators of the form f ⋆ S for a trace-class operator S. The results of this paper indicate that these operators describe the time-frequency localization in various ways. For example we are interested in the amount of "spreading" in time and frequency that an operator performs on a function which we describe in form of bounds on the concentration of the spreading function, or equivalently ...

show abstract

“…For such states L(ρ) is well-defined and trace class (see e.g. Propositions 3.14 and 3.15 of [26]), and using the non-commutative Parseval relation, Theorem 1, we can write the Dirichlet form (defined in Equation (3.11)) as follows:…”

Section: Proofs Of Theorem and Theoremmentioning

confidence: 99%

Contractivity properties of a quantum diffusion semigroup

Datta

Pautrat

Rouzé

2017

Journal of Mathematical Physics

View full text Add to dashboard Cite

We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

show abstract

Schwartz operators

Cited by 39 publications

References 77 publications

Convolutions for localization operators

Convolutions for localization operators

Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

Contractivity properties of a quantum diffusion semigroup

Contact Info

Product

Resources

About