Ortega-Cerdà-Seip demonstrated that there are bounded multiplicative Hankel forms which do not arise from bounded symbols. On the other hand, when such a form is in the Hilbert-Schmidt class ₂, Helson showed that it has a bounded symbol. The present work investigates forms belonging to the Schatten classes between these two cases. It is shown that for every $p>{(1-log\pi /log4)}^{-1}$ there exist multiplicative Hankel forms in the Schatten class ${}_p$ which lack bounded symbols. The lower bound on p is in a certain...

We consider the solution operator $S{ℱ}_{\mu ,(p,q)}\to L^2{\left(\mu \right)}_{(p,q)}$ to the $\overline{\partial }$-operator restricted to forms with coefficients in ${ℱ}_{\mu }=\left\{ff\text{is}\text{entire}\text{and}{\int }_{{ℂ}^n}{\left|f\left(z\right)\right|}^2\mathrm{d}\mu \left(z\right)<\infty \right\}$. Here ${ℱ}_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in ${ℱ}_{\mu }$, $L^2\left(\mu \right)$ is the corresponding $L^2$-space and $\mu$ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\overline{\partial }$. This solution operator will have the property $Sv\perp {ℱ}_{(p,q)}\forall v\in {ℱ}_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness...

In this paper, we obtain the Fekete-Szego inequalities for the functions of complex order defined by convolution. Also, we find upper bounds for the second Hankel determinant $|a_2{a}_4-a_3^2|$ for functions belonging to the class $S_{\gamma }^b(g\left(z\right);A,B)$.

In this paper the notion of slant Hankel operator $K_{\varphi }$, with symbol $\varphi$ in $L^{\infty }$, on the space $L^2\left(𝕋\right)$, $𝕋$ being the unit circle, is introduced. The matrix of the slant Hankel operator with respect to the usual basis $\{z^i:i\in \mathbb{Z}\}$ of the space $L^2$ is given by $\langle {\alpha }_{ij}\rangle =\langle a_{-2i-j}\rangle$, where $\sum _{i=-\infty }^{\infty }{a}_i{z}^i$ is the Fourier expansion of $\varphi$. Some algebraic properties such as the norm, compactness of the operator $K_{\varphi }$ are discussed. Along with the algebraic properties some spectral properties of such operators are discussed. Precisely, it is proved that for...

We consider a class of unbounded nonhyperbolic complete Reinhardt domains $$D_{n,m,k}^{\mu ,p,s}:=\left\{(z,w_1,\cdots ,w_m)\in {ℂ}^n\times {ℂ}^{k_1}\times \cdots \times {ℂ}^{k_m}:\frac{\parallel w_1{\parallel }^{2p_1}}{{\mathrm{e}}^{-{\mu }_1{\parallel z\parallel }^s}}+\cdots +\frac{\parallel w_m{\parallel }^{2p_m}}{{\mathrm{e}}^{-{\mu }_m{\parallel z\parallel }^s}}<1\right\},$$ where $s$, $p_1,\cdots ,p_m$, ${\mu }_1,\cdots ,{\mu }_m$ are positive real numbers and $n$, $k_1,\cdots ,k_m$ are positive integers. We show that if a Hankel operator with anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $A^2\left(D_{n,m,k}^{\mu ,p,s}\right)$, then it must be zero. This gives an example of high dimensional unbounded complete Reinhardt domain that does not admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.

We present sufficient conditions for the existence of $p$th powers of a quasihomogeneous Toeplitz operator $T_{{\mathrm{e}}^{\mathrm{i}s\theta }\psi }$, where $\psi$ is a radial polynomial function and $p$, $s$ are natural numbers. A large class of examples is provided to illustrate our results. To our best knowledge those examples are not covered by the current literature. The main tools in the proof of our results are the Mellin transform and some classical theorems of complex analysis.

The objective of this paper is to obtain sharp upper bound for the function $f$ for the second Hankel determinant $|a_2{a}_4-a_3^2|$, when it belongs to the class of functions whose derivative has a positive real part of order $\alpha$ $(0\le \alpha <1)$, denoted by $RT\left(\alpha \right)$. Further, an upper bound for the inverse function of $f$ for the nonlinear functional (also called the second Hankel functional), denoted by $|t_2{t}_4-t_3^2|$, was determined when it belongs to the same class of functions, using Toeplitz determinants.

The purpose of this paper is to study the Sarason’s problem on Fock spaces of polyanalytic functions. Namely, given two polyanalytic symbols $f$ and $g$, we establish a necessary and sufficient condition for the boundedness of some Toeplitz products $T_f{T}_{\overline{g}}$ subjected to certain restriction on $f$ and $g$. We also characterize this property in terms of the Berezin transform.

We present here a quite unexpected result: If the product of two quasihomogeneous Toeplitz operators $T_f{T}_g$ on the harmonic Bergman space is equal to a Toeplitz operator $T_h$, then the product $T_g{T}_f$ is also the Toeplitz operator $T_h$, and hence $T_f$ commutes with $T_g$. From this we give necessary and sufficient conditions for the product of two Toeplitz operators, one quasihomogeneous and the other monomial, to be a Toeplitz operator.