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Continuity and compositions of operators with kernels in ultra-test function and ultra-distribution spaces
Linnaeus University, Faculty of Technology, Department of Mathematics.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we consider continuity and positivity properties of pseudo-differential operators in Gelfand-Shilov and Pilipović spaces, and their distribution spaces. We also investigate composition property of pseudo-differential operators with symbols in quasi-Banach modulation spaces.

We prove that positive elements with respect to the twisted convolutions, possesing Gevrey regularity of certain order at origin, belong to the Gelfand-Shilov space of the same order. We apply this result to positive semi-definite pseudo-differential operators, as well as show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals.

We also prove that any linear operator with kernel in a Pilipović or Gelfand-Shilov space can be factorized by two operators in the same class. We give links on numerical approximations for such compositions and apply these composition rules to deduce estimates of singular values and establish Schatten-von Neumann properties for such operators.  

Furthermore, we derive sufficient and necessary conditions for continuity of the Weyl product with symbols in quasi-Banach modulation spaces.

Place, publisher, year, edition, pages
Växjö: Linnaeus University Press, 2016. , 178 p.
Series
Linnaeus University Dissertations, 263/2016
Keyword [en]
Composition, modulation spaces, positivity, pseudo-differential operators, Schatten-von Neumann operators, twisted convolutions, ultra-distributions
National Category
Mathematics
Research subject
Mathematics, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-58076ISBN: 978-91-88357-38-0OAI: oai:DiVA.org:lnu-58076DiVA: diva2:1045986
Public defence
2016-11-17, C1202, Hus C, Växjö, 13:15 (English)
Opponent
Supervisors
Available from: 2016-11-14 Created: 2016-11-11 Last updated: 2016-11-22Bibliographically approved
List of papers
1. Boundedness of Gevrey and Gelfand-Shilov kernels of positive semi-definite operators
Open this publication in new window or tab >>Boundedness of Gevrey and Gelfand-Shilov kernels of positive semi-definite operators
2015 (English)In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 6, no 2, 153-185 p.Article in journal (Refereed) Published
Abstract [en]

We show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals. We also prove that positive elements with respect to the twisted convolution, belonging to a Gevrey class of certain order at the origin, belong to the Gelfand-Shilov space of the same order. In the end we apply these results to positive semi-definite pseudo-differential operators.

Keyword
Positivity, Twisted convolutions, Ultra-distributions, Weyl quantization, Pseudo-differential operators
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-46283 (URN)10.1007/s11868-015-0116-x (DOI)000355234500001 ()
Available from: 2015-09-14 Created: 2015-09-14 Last updated: 2016-11-11Bibliographically approved

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