[Turkmath:4730] İ.Ü. Matematik Bölümü Semineri

[TEMHA ERKOÇ YILMAZTÜRK]

Merhabalar,

06.01.2021 tarihinde  saat 13.15 te   Prof. Joachim Toft  (Linnæus
University, Sweden) başlık ve özeti aşağıda verilen bir konuşma yapacaktır.

Seminer Zoom programı üzerinden online yapılacaktır. Katılmak isteyenlerin
katılım bilgilerini alabilmeleri için "huseyinuysal at istanbul.edu.tr "
adresine mail atmaları gerekmektedir.

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*Başlık :*  Analytic pseudo-differential calculus via the Bargmann transform


*Özet:*  The Bargmann transform is a transform which maps Fourier-invariant
function spaces and their duals to certain spaces of formal power series
expansions, which sometimes are convenient classes of analytic functions.
In the 70th, Berezin used the Bargmann transform to translate problems in
operator theory into an analytic pseudo-differential calculus, the
so-called Wick calculus, where the involved symbols are analytic functions,
and the corresponding operators map suitable classes of entire functions
into other classes of entire functions. In the same manner, the Toeplitz
operators correspond to so-called anti-Wick operators on the Bargmann
transformed side.
Recently, some investigations on certain Fourier invariant subspaces of the
Schwartz space and their dual (distribution) spaces have been performed by
the author. These spaces are called Pilipovi ́c spaces, and are defined by
imposing suitable boundaries on the Hermite coefficients of the involved
functions or distributions. The family of Pilipovi ́c spaces contains all
Fourier invariant Gelfand-Shilov spaces as well as other spaces which are
strictly smaller than any Fourier invariant non-trivialGelfand-Shilov
space. In the same way, the family of Pilipovi ́c distribution spaces
contains spaces which are strictly larger than any Fourier invariant
Gelfand-Shilov distribution space.
In the talk we show that the Bargmann images of Pilipovi ́c spaces and
their distribution spaces are convenient classes of analytic functions or
power series expansions which are suitable when investigating Wick
operators (i. e. the operators in the Wick calculus).
We deduce continuity properties for such operators when the symbols and
target functions possess certain (weighted) Lebesgue estimates. We also
explain how the counter images with respect to the Bargmann transform of
these results generalise some continuity results for (real)
pseudo-differential operators with symbols in modulation spaces, when
acting on other modulation space. Finally we discuss some links between
ellipticity in the real pseudo-differential calculus and the Wick calculus,
as well as links between Wick and anti-Wick operators.
The talk is based on collaborations with Nenad Teofanov and Patrik
Wahlberg, and parts of the content of the talk is available at:

N. Teofanov, J. Toft Pseudo-differential calculus in a Bargmann setting,
Ann. Acad. Sci. Fenn.
Math. 45 (2020), 227–257.
N. Teofanov, J. Toft, P. Wahlberg Pseudo-differential operators with
isotropic symbols, and Wick and anti-Wick operators, arXiv:??


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İyi günler dileğiyle,

Temha

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