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Register a new userNeharis theorem for convex domain Hankel and Toeplitz operators in several variables
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We prove Neharis theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows. Let $Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2Xi$, consider the Hankel operator $$Gamma_f (g)(x)=int_{Xi} f(x+y) g(y) , dy, quad x inXi.$$ Then $Gamma_f$ extends to a bounded operator on $L^2(Xi)$ if and only if there is a bounded function $b$ on $mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2Xi$. This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix. In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.
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In this paper, we investigate the boundedness of Toeplitz product $T_{f}T_{g}$ and Hankel product $H_{f}^{*} H_{g}$ on Fock-Sobolev space for two polynomials $f$ and $g$ in $z,overline{z}inmathbb{C}^{n}$. As a result, the boundedness of Toeplitz operator $T_{f}$ and Hankel operator $H_{f}$ with the polynomial symbol $f$ in $z,overline{z}inmathbb{C}^{n}$ is characterized.
We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
We obtain asymptotics for Toeplitz, Hankel, and Toeplitz+Hankel determinants whose symbols possess Fisher-Hartwig singularities. Details of the proofs will be presented in another publication.
For $-1<alpha<infty$, let $omega_alpha(z)=(1+alpha)(1-|z|^2)^alpha$ be the standard weight on the unit disk. In this note, we provide descriptions of the boundedness and compactness for the Toeplitz operators $T_{mu,beta}$ between distinct weighted Bergman spaces $L_{a}^{p}(omega_{alpha})$ and $L_{a}^{q}(omega_{beta})$ when $0<pleq1$, $q=1$, $-1<alpha,beta<infty$ and $0<pleq 1<q<infty, -1<betaleqalpha<infty$, respectively. Our results can be viewed as extensions of Pau and Zhaos work in cite{Pau}. Moreover, partial of main results are new even in the unweighted settings.
Suppose that $phi$ and $psi$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_phi$ and $T_psi$ denote the Toeplitz operators with symbols $phi$ and $psi$ respectively. We give an explicit formula for the determinant of $T_phi T_psi T_phi^{-1} T_psi^{-1}$ in terms of the products of the tame symbols of $phi$ and $psi$ on the open unit disk.
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