Let $\mu$ be a finite positive measure on the unit disk and let $j\ge 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{\mu }^{\left(j\right)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class for $1\le p<\infty$ on the Bergman space $A^2$, and then give a sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class $(0<p<1)$ on $A^2$. We also discuss the generalized Toeplitz operators with general bounded symbols. If $\varphi \in L^{\infty }(D,\mathrm{d}A)$ and $1<p<\infty$, we define the generalized...

Let $A_{\zeta }=\Omega -\overline{\rho \left(\zeta \right)·\Omega }$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{\zeta }\left(z\right)$ of $A_{\zeta }$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{\zeta }$ is non-pseudoconvex when the dimension of $A_{\zeta }$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ²log{K}_{\zeta }/\partial \zeta \partial \zeta ̅$, as well as its boundary behavior.

We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on the weighted Bergman space $A^2\left(\Omega \right)$, where $\Omega$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^p{\overline{z}}^q}$ to be complex symmetric. When $p,q\in {\mathbb{N}}^2$, we prove that $T_{z^p{\overline{z}}^q}$ is complex symmetric on $A^2\left(\Omega \right)$ if and only if $p_1=q_2$ and $p_2=q_1$. Moreover, we completely characterize when monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on $A^2\left({𝔻}_n\right)$ are $J_U$-symmetric with the $n\times n$ symmetric unitary matrix $U$.

Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping $h$ and that of $zh$, we study various $L^2$ norms for $T_{\varphi }\left(h\right)$, where $T_{\varphi }$ is the Toeplitz operator with symbol $\varphi$. In Theorem , given polynomials $p$ and $q$ we find a symbol $\varphi$ such that $T_{\varphi }\left(p\right)=q$. We extend some of our results to the polydisc.

Let $\mu$ be a positive Borel measure on the complex plane ${ℂ}^n$ and let $j=(j_1,\cdots ,j_n)$ with $j_i\in \mathbb{N}$. We study the generalized Toeplitz operators $T_{\mu }^{\left(j\right)}$ on the Fock space $F_{\alpha }^2$. We prove that $T_{\mu }^{\left(j\right)}$ is bounded (or compact) on $F_{\alpha }^2$ if and only if $\mu$ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class for $1\le p<\infty$.

Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by $T_{z,p}\left(f\right)={(1-|zₙ\left|²\right)}^{1/p}f\left(zₙ\right)$. Necessary and sufficient conditions on zₙ are given such that $T_{z,p}$ maps the Hardy space $H^p$ boundedly into the sequence space ${\ell }^q$. A corresponding result for Bergman spaces is also stated.

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.

Let μ be a finite positive Borel measure on [0,1). Let ${\mathscr{H}}_{\mu }={\left({\mu }_{n,k}\right)}_{n,k\ge 0}$ be the Hankel matrix with entries ${\mu }_{n,k}={\int }_{[0,1)}{t}^{n+k}d\mu \left(t\right)$. The matrix ${}_{\mu }$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula ${\mathscr{H}}_{\mu }\left(f\right)\left(z\right)={\sum }_{n=0}^{\infty }i\left({\sum }_{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)zⁿ$, z ∈ , where $f\left(z\right)={\sum }_{n=0}^{\infty }aₙzⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that ${\mathscr{H}}_{\mu }\left(f\right)\left(z\right)={\int }_{[0,1)}f\left(t\right)/(1-tz)d\mu \left(t\right)$ for all f in the Hardy space H¹, and among them we describe those for which ${\mathscr{H}}_{\mu }$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A². ...

Let $u$ be a holomorphic function and $\varphi$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u{C}_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$, $\varphi$, their derivatives, and ${\varphi }^n$, the $n$-th power of $\varphi$. Moreover, we obtain some similar estimates for the essential norms of the operators $u{C}_{\varphi }$, from which sufficient and necessary conditions of compactness of $u{C}_{\varphi }$ follows immediately. ...

Let $u$ be an inner function and $K_u^2$ be the corresponding model space. For an inner function $v$, the subspace $v{H}^2$ is an invariant subspace of the unilateral shift operator on $H^2$. In this article, using the structure of a Toeplitz kernel $\ker T_{\overline{u}v}$, we study the intersection $K_u^2\cap v{H}^2$ by properties of inner functions $u$ and $v$ $(v\ne u)$. If $K_u^2\cap v{H}^2\ne \left\{0\right\}$, then there exists a triple $(B,b,g)$ such that $$\overline{u}v=\frac{\lambda b\overline{B{O}_g}}{g},$$ where the triple $(B,b,g)$ means that $B$ and $b$ are Blaschke products, $g$ is an invertible function in $H^{\infty }$, $O_g$ denotes the outer factor...