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Register a new userProducts of Toeplitz and Hankel Operators on Fock-Sobolev Spaces
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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.
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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.
A multiplicative Hankel operator is an operator with matrix representation $M(alpha) = {alpha(nm)}_{n,m=1}^infty$, where $alpha$ is the generating sequence of $M(alpha)$. Let $mathcal{M}$ and $mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(alpha) in mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(beta) in mathcal{M}_0$, $$|M(alpha) - N(beta)|_{mathcal{B}(ell^2(mathbb{N}))} = inf left {|M(alpha) - K |_{mathcal{B}(ell^2(mathbb{N}))} , : , K colon ell^2(mathbb{N}) to ell^2(mathbb{N}) textrm{ compact} right}.$$ Intimately connected with this result, it is then proven that the bidual of $mathcal{M}_0$ is isometrically isomorphic to $mathcal{M}$, $mathcal{M}_0^{ast ast} simeq mathcal{M}$. It follows that $mathcal{M}_0$ is an M-ideal in $mathcal{M}$. The dual space $mathcal{M}_0^ast$ is isometrically isomorphic to a projective tensor product with respect to Dirichlet convolution. The stated results are also valid for small Hankel operators on the Hardy space $H^2(mathbb{D}^d)$ of a finite polydisk.
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.
We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<infty$ of the Hankel operators $H_ f$ and $H_{bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.
For a pointwise multiplier $varphi$ of the Hardy-Sobolev space $H^2_beta$ on the open unit ball $bn$ in $cn$, we study spectral properties of the multiplication operator $M_varphi: H^2_betato H^2_beta$. In particular, we compute the spectrum and essential spectrum of $M_varphi$ and develop the Fredholm theory for these operators.
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