Schatten Classes for Toeplitz Operators with Hilbert Space Windows on Modulation Spaces.

The aim of this paper is to study Schatten–von Neumann properties of Toeplitz operators with Hilbert windows, acting on Hilbert modulation spaces, i.e. modulation spaces of Hilbert type. Hee and Wong investigated Schatten–von Neumann properties for Toeplitz operators in a powerful way from an abstract point of view. Their results in the general frame of square integrable representations on a Hilbert space are systemized in a some papers which have inspired our work. Actually we shall be especially concerned with various extensions of some of these results. Thereafter we apply our Toeplitz results to prove Young type properties for convolutions between weighted Lebesgue spaces and Schatten–von Neumann classes of symbols in pseudo-differential calculus. Toeplitz operators, or localization operators, appear in the literature in many different situations. They were introduced in time-frequency analysis by Daubechies as certain filters for signals. Lieb and Solovej reformulated problems in quantum mechanics in terms of coherent state transforms, where Toeplitz operators appeared in a natural way. Since then, boundedness and compactness properties of such operators have been investigated in many works. Many of the problems concern Toeplitz operators when acting on Hilbert spaces, especially on L2. In the present paper we consider such questions when the Hilbert spaces are particular classes of modulation spaces and the symbols belong to weighted Lebesgue spaces. Modulation spaces have shown to be a natural class of Banach spaces in certain parts of Fourier analysis, e.g. time-frequency analysis, where the modulation space norms can be used to measure the time-frequency content of signals. At the same time they generalize various Sobolev-type spaces, and in some problems they are appropriate when considering regularity and decay properties of involved functions and distributions. Finally we note that there are many different characterizations available for modulation spaces. For example, they are easy to discretize by means of Gabor frames. In certain situations they can also be characterized in terms of Toeplitz operators. These properties also lead to the fact that modulation spaces have been useful when discussing certain problems of localization operators and pseudo-differential operators, which in some sense can be considered as a part of Fourier Analysis. Actually in the final sections of the paper Schatten–von Neumann symbols of operators acting on Hilbert modulation spaces are used to prove certain composition results for Weyl symbols of trace class characters with analytic functions. In these investigations we use the fact that the Weyl symbol of the Toeplitz operator can be expressed as a convolution between the Toeplitz symbol and a Wigner distribution.