Sarason’s Conjecture of Toeplitz Operators on Fock-Sobolev Type Spaces

Xiaofeng Wang,Jianjun Chen,Jin Xia

doi:10.1155/2018/1571365

Xiaofeng Wang, Jianjun Chen + Show 1 more

Open Access

https://doi.org/10.1155/2018/1571365

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Abstract

In this note, we will solve Sarason’s conjecture on the Fock-Sobolev type spaces and give a well solution that if Toeplitz product TuTv¯, with entire symbols u and v, is bounded if and only if u=eq, v=Ce-q, where q is a linear complex polynomial and C is a nonzero constant.

Highlights

Let Cn denote the complex n-space and dV be the ordinary volume measure on Cn that is normalized so that∫Cn e−|z|2 dV(z) = 1
If we give that u and V are two nonzero functions in the Fock-Sobolev type space Fα2 such that Toeplitz product TuαTVα is polynomial q(z) bounded on on Cn such
If u and V are two functions in the Fock-Sobolev type space Fα2, not identically zero, such that the operator TuαTVα is bounded on Fα2, |̃ u|2(z)|̃ V|2(z) is a bounded function on the complex space

Summary

Introduction

In the case of the Fock-Sobolev space, Chen et al (see [12]) had already proven the same topics and obtained the similar results What is more, they claimed that if f and g are two nonzero functions in the Fock(-Sobolev) space, the Toeplitz product TfTg is bounded if and only if f = eq and g = Ce−q, where C is a nonzero constant and q is a linear polynomial. Bommier-Hato et al in [14] continued to research Cho’s results on the general Fock-type space with the weight functions exp(−| ⋅ |2m) They took full advantage of the exact form of the reproducing kernel of the general Fock-type space and concluded that if u and V are two nonzero functions, the Toeplitz product TuTV is bounded if and only if u = eg and V = Ce−g, where C is a nonzero constant and g is a polynomial of degree at most m.

Proof of the Main Result

Conclusions