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.

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 $\varphi$ be a substitution of Pisot type on the alphabet $𝒜=\{1,2,...,d\}$; $\varphi$ satisfies theif for every $i,j\in 𝒜$, there are integers $k,n$ such that ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same $k$-th letter, and the prefixes of length $k-1$ of ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d=2$ and provide a partial result for $d\ge 2$.

Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega ⋐X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega$ is a weakly $q$-pseudoconvex domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\overline{\partial }$ on $\Omega$. Moreover, we study the $\overline{\partial }$-problem on $\Omega$. Specifically, we use the modified weight function method to study the weighted $\overline{\partial }$-problem with exact support in $\Omega$. Our method relies on the $L^2$-estimates by Hörmander (1965) and by Kohn (1973). ...

We prove that, assuming , if $X$ is a space with ${\aleph }_1$-calibre and a zeroset diagonal, then $X$ is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular $G_{\delta }$-diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with ${\aleph }_1$-calibre.

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...