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 Ufimsk. Mat. Zh., 2018, Volume 10, Issue 4, Pages 85–91 (Mi ufa450)

On some linear operators on Fock type space

I. Kh. Musin

Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevskii str. 112, 450077, Ufa, Russia

Abstract: We consider a lower semi-continuous function $\varphi$ in $\mathbb{R}^n$ depending on the absolute values of the variables and growing faster than $a \ln (1 + \Vert x \Vert)$ for each positive $a$. In terms of this function, we define a Hilbert space $F^2_{\varphi}$ of entire functions in $\mathbb{C}^n$. This is a natural generalization of a classical Fock space. In this paper we provide an alternative description of the space $F^2_{\varphi}$ in terms of the coefficients in the power expansions for the entire functions in this space. We mention simplest properties of reproducing kernels in the space $F^2_{\varphi}$. We consider the orthogonal projector from the space $L^2_{\varphi}$ of measurable complex-valued functions $f$ in $\mathbb{C}^n$ such that
$$\Vert f \Vert_{\varphi}^2 = \int_{\mathbb{C}^n} \vert f(z)\vert^2 e^{- 2 \varphi (\mathrm{abs} z)} d \mu_n (z) < \infty ,$$
where $z =(z_1, \ldots , z_n)$, $\mathrm{abs} z = (\vert z_1 \vert, \ldots , \vert z_1 \vert)$, on its closed subspace $F^2_{\varphi}$, and for this projector we obtain an integral representation.
We also obtain an integral formula for the trace of a positive linear continuous operator on the space $F^2_{\varphi}$. By means of this formula we find the conditions, under which a weighted operator of the composition on $F^2_{\varphi}$ is a Hilbert–Schmidt operator. Two latter results generalize corresponding results by Sei-Ichiro Ueki, who studied similar questions for operators in Fock space.

Keywords: entire functions, Fock type space, linear operators, operator trace, weighted composition operators, Hilbert–Schmidt operator.

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English version:
Ufa Mathematical Journal, 2018, 10:4, 85–91 (PDF, 309 kB); https://doi.org/10.13108/2018-10-4-85

Bibliographic databases:

UDC: 517.555
MSC: 32A15, 42B10, 46E22, 47B33

Citation: I. Kh. Musin, “On some linear operators on Fock type space”, Ufimsk. Mat. Zh., 10:4 (2018), 85–91; Ufa Math. J., 10:4 (2018), 85–91

Citation in format AMSBIB
\Bibitem{Mus18} \by I.~Kh.~Musin \paper On some linear operators on Fock type space \jour Ufimsk. Mat. Zh. \yr 2018 \vol 10 \issue 4 \pages 85--91 \mathnet{http://mi.mathnet.ru/ufa450} \transl \jour Ufa Math. J. \yr 2018 \vol 10 \issue 4 \pages 85--91 \crossref{https://doi.org/10.13108/2018-10-4-85} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457367000008}