We study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols $f$ and $g$, we show that $T_fT_g$ is compact if and only if $fg$ vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of $fg$ on the whole polydisc might not imply the compactness of $T_fT_g$. On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.

We focus on two problems relating to the question of when the product of two posinormal operators is posinormal, giving (1) necessary conditions and sufficient conditions for posinormal operators to have closed range, and (2) sufficient conditions for the product of commuting closed-range posinormal operators to be posinormal with closed range. We also discuss the relationship between posinormal operators and EP operators (as well as hypo-EP operators), concluding with a new proof of the Hartwig-Katz Theorem, which characterizes when the product of posinormal operators on $\mathbb{C}^n$ is posinormal.

We find sufficient conditions for a self-map of the unit ball to converge uniformly under iteration to a fixed point or idempotent on the entire ball. Using these tools, we establish spectral containments for weighted composition operators on Hardy and Bergman spaces of the ball. When the compositional symbol is in the Schur-Agler class, we establish the spectral radii of these weighted composition operators.

Given a weighted shift $T$ of multiplicity two, we study the set $\sqrt{T}$ of all square roots of $T$. We determine necessary and sufficient conditions on the weight sequence so that this set is non-empty. We show that when such conditions are satisfied, $\sqrt{T}$ contains a certain special class of operators. We also obtain a complete description of all operators in $\sqrt{T}$.

In this paper we generalize the classical theorems of Brown and Halmos about algebraic properties of Toeplitz operators to the Bergman spaces over the unit ball in several complex variables. A key result, which is of independent interest, is the characterization of summable functions $u$ on the unit ball whose Berezin transform $B(u)$ can be written as a finite sum $\sum_{j}f_j\,\bar{g}_j$ with all $f_j, g_j$ being holomorphic. In particular, we show that such a function must be pluriharmonic if it is sufficiently smooth and bounded. We also settle an open question about $\mathcal{M}$-harmonic functions. Our proofs employ techniques and results from function and operator theory as well as partial differential equations.

We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called Shapiro--Shields functions andshift-invariant subspaces generated by polynomials. Applying our results, we show that the only entire inner functions on weighted Hardy spaces over the unit disk are multiples of monomials, extending recent work of Cobos and Seco.

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution $\diamond$ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if $F_j, G_j$ ($1\leq j\leq N$) are polynomials of $z$ and $\bar{z}$ then $\sum_{j=1}^{N}T_{F_j}T_{G_j}-T_{H}$ is a finite rank operator for some $L^{1}$-function $H$ if and only if $\sum_{j=1}^{N}F_j\diamond G_j$ belongs to $L^1$ and $H=\sum_{j=1}^{N}F_j\diamond G_j$. In the case $F_j$'s are holomorphic and $G_j$'s are conjugate holomorphic, it is shown that $H$ is a solution to a system of first order partial differential equations with a constraint.

We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix. We apply this result to the Toeplitz operator with specific algebraic symbols acting on certain weighted Bergman spaces of the unit disk and determine when such operators are hyponormal.

Inner functions play a central role in function theory and operator theory on the Hardy space over the unit disk. Motivated by recent works of C. Beneteau et al. and of D. Seco, we discuss inner functions on more general weighted Hardy spaces and investigate a method to construct analogues of finite Blaschke products.

We show that any $m$-isometric tuple of commuting algebraic operators on a Hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple. Our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables.

Chu and Khavinson recently obtained a lower bound for the norm of the self-commutator of a certain class of hyponormal Toeplitz operators on the Hardy space. Via a different approach, we offer a generalization of their result.

It is already known that the Cesaro matrices of orders one and two are coposinormal operators on $\ell^2$.Here it is shown that the Cesaro matrices of all orders are coposinormal; the proof employs posinormality, achieved by means of a diagonal interrupter, and makes use of the Zeilberger's algorithm and computational assistance by Maple.

We make a progress towards describing the commutants of Toeplitz operators with harmonic symbols on the Bergman space over the unit disk. Our work greatly generalizes several partial results in the field.

We study composition operators acting on Hilbert spaces of entire functions in several variables. Depending on the defining weight sequence of the space, different criteria for boundedness and compactness are developed. Our work extends several known results on Fock spaces and other spaces of entire functions.

If $\varphi$ is a bounded separately radial function on the unit ball, the Toeplitz operator $T_{\varphi}$ is diagonalizable with respect to the standard orthogonal basis of monomials on the Bergman space. Given such a $\varphi$, we characterize bounded functions $\psi$ for which $T_{\psi}$ commutes with $T_{\varphi}$. Several examples are given to illustrate our results.

For an arbitrary Hilbert space $E$, the Segal--Bargmann space $\mathcal{H}(E)$ is the reproducing kernel Hilbert space associated with the kernel $K(x,y)=\exp(\langle x,y\rangle)$ for $x,y$ in $E$. If $\varphi:E_1\rightarrow E_2$ is a mapping between two Hilbert spaces, then the composition operator $C_{\varphi}$ is defined by $C_{\varphi}h = h\circ\varphi$ for all $h\in \mathcal{H}(E_2)$ for which $h\circ\varphi$ belongs to $\mathcal{H}(E_1)$. We determine necessary and sufficient conditions for the boundedness and compactness of $C_{\varphi}$. In the special case where $E_1=E_2=\mathbb{C}^n$, we recover results obtained by Carswell, MacCluer and Schuster. We also compute the spectral radii and the essential norms of a class of operators $C_{\varphi}$.

We study composition operators acting between $\mathcal{N}_p$-spaces in the unit ball in $\mathbb{C}^m$. We obtain characterizations of the boundedness and compactness of $C_{\phi}:\mathcal{N}_p\rightarrow \mathcal{N}_q$ for $p, q>0$.

It is well known that on the Hardy space $H^2(\mathbb{D})$ or weighted Bergman space $A^2_{\alpha}(\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces.

We obtain simple characterizations of unilateral and bilateral weighted shift operators that are $m$-isometric. We show that any such operator is a Hadamard product of $2$-isometries and $3$-isometries. We also study weighted shift operators whose powers are $m$-isometric.

We study the commuting problem of Toeplitz operators on the Fock space over $\mathbb{C}^n$. Given a separately radial polynomial $\varphi$ in $z$ and $\bar{z}$, we characterize polynomially bounded functions $\psi$ such that the operators $T_{\psi}$ and $T_{\varphi}$ commute. Several examples and consequences are discussed.

Properties of $m$-selfadjoint and $m$-isometric operators have been investigated by several researchers. Particularly interesting to us are algebraic properties of nilpotent perturbations of such operators. McCullough and Rodman showed in the nineties that if $Q^n=0$ and $A$ is a selfadjoint operator commuting with $Q$ then the sum $A+Q$ is a $(2n-1)$-selfadjoint operator. Very recently, Bermudez, Martinon, and Noda proved a similar result for nilpotent perturbations of isometries.Via a new approach, we obtain simple proofs of these results and other generalizations to operator roots of polynomials.

In this paper, we study the product of a composition operator $C_{\varphi}$ with the adjoint of a composition operator $C_{\psi}^{*}$ on the Hardy space $H^2$.The order of the product gives rise to two different cases.We completely characterize when the operator $C_{\varphi}C_{\psi}^{*}$ is invertible, isometric, and unitary and when the operator $C_{\psi}^{*}C_{\varphi}$ is isometric and unitary. If one of the inducing maps $\varphi$ or $\psi$ is univalent, we completely characterize when$C_{\psi}^{*}C_{\varphi}$ is invertible.

We obtain new and simple characterizations for the boundedness and compactness of weighted composition operators on the Fock space over $\mathbb{C}$. We also describe all weighted composition operators that are normal or isometric.

Let $\mathcal{R}$ be an arbitrary bounded complete Reinhardt domain in $\mathbb{C}^n$. We show that for $n\geq 2$, if a Hankel operator with an anti-holomorphic symbol is Hilbert--Schmidt on the Bergman space $A^2(\mathcal{R})$, then it must equal zero. This fact has previously been proved only for strongly pseudoconvex domains and for a certain class of pseudoconvex domains.

Motivated by the work of Nazarov and Shapiro on the unit disk, we study asymptotic Toeplitzness of composition operators on the Hardy space of the unit sphere in $\mathbb{C}^n$. We extend some of their results but we also show that new phenomena appear in higher dimensions.

We study algebraic properties of Toeplitz operators on Bergman spaces of polyanalytic functions on the unit disk. We obtain results on finite-rank commutators and semi-commutators of Toeplitz operators with harmonic symbols. We also raise and discuss some open questions.

We study Toeplitz operators with uniformly continuous symbols on generalized harmonic Bergman spaces of the unit ball in ${R}^n$. We describe their essential spectra and establish a short exact sequence associated with the $C^{*}$-algebra generated by these operators.

We study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form $(1~-~\langle z,w\rangle)^{-\gamma}$ for $\gamma,0$. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weighted composition operators. Normality of these operators is also investigated.

We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in $\mathbb{C}^n$. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class $Sym_{,0}(\mathbb{C}^n)$ of symbols having certain growth at infinity. We then provide explicit examples of zero products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal-Bargmann space and on the Bergman space over the unit ball.Finally, we discuss the finite rank problem. We show that there are no non-trivial rank one Toeplitz operators $T_f$ for certain classes of $f$. In all these problems, the growth at infinity of the symbols plays a crucial role.

We consider Toeplitz operators with symbols enjoying a uniform radial limit on Segal-Bargmann type spaces. We show that such an operator is compact if and only if the limiting function vanishes on the unit sphere. The structure of the $C^{*}$-algebra generated by Toeplitz operators whose symbols admit continuous uniform radial limits is also analyzed.

For $\alpha>-1$, let $A^{2}_{\alpha}$ denote the corresponding weighted Bergman space of the unit ball. For any self-adjoint subset $G\subset L^{\infty}$, let ${T}(G)$ denote the $C^{*}$-algebra generated by $\{T_{f}: f\in G\}$. Let ${CT}(G)$ denote the commutator ideal of ${T}(G)$. It was showed by D. Suarez (in 2004 for $n=1$) and by the author (in 2006 for all $n\geq 1$) that ${CT}(L^{\infty})={T}(L^{\infty})$ in the case $\alpha=0$. In this paper we show that in the setting of weighted Bergman spaces, the identity ${CT}(G)={T}(G)$ holds true for a class of subsets $G$ including $L^{\infty}$.

Let $\vartheta$ be a measure on the polydisc $D^n$ which is the product of $n$ regular Borel probability measures so that $\vartheta([r,1)^n\times T^n),0$ for all $r\in (0,1)$. The Bergman space $A^2_{\vartheta}$ consists of all holomorphic functions that are square integrable with respect to $\vartheta$. In one dimension, it is well known that if $f$ is continuous on the closed disc $\overline{D}$, then the Hankel operator $H_{f}$ is compact on $A^2_{\vartheta}$. In this paper we show that for $n\geq 2$ and $f$ a continuous function on $\overline{D}^n$, $H_{f}$ is compact on $A^2_{\vartheta}$ if and only if there is a decomposition $f=h+g$, where $h$ belongs to $A^2_{\vartheta}$ and $\lim_{z\to\partial D}g(z)=0$.

For a bounded measurable function $f$ on the open unit disk $ D$, let $T_f$ denote the corresponding Toeplitz operator on the Bergman space $A^2( D)$. A recent result of D. Luecking shows that if $T_f$ has finite rank, then $f$ must be the zero function. Using a refined version of this result, we show that if all, except possibly one, of the functions $f_1,\ldots, f_{m}$ are radial and $T_{f_1}\cdots T_{f_m}$ has finite rank, then one of these functions must be zero.

If $T= \big[ T_1 \ldots T_n\big]$ is a row contraction with commuting entries, and the Arveson dilation is $\tilde{T}= \big[\tilde{T}_1 \ldots \tilde{T}_n\big]$, then any operator $X$ commuting with each $T_i$ dilates to an operator $Y$ of the same norm which commutes with each $\tilde{T}_i$.

We study Toeplitz operators on the Bergman space $A^2_{\vartheta}$ of the unit polydisk $ D^n$, where $\vartheta$ is a product of $n$ rotation-invariant regular Borel probability measures. We show that if $f$ is a bounded Borel function on $ D^n$ such that $F(w)=\lim_{z\rightarrow w}f(z)$ exists for all $w\in\partial D^n$, then $T_f$ is compact if and only if $F=0$ a.e. with respect to a measure $\gamma$ associated with $\vartheta$ on the boundary $\partial D^n$ . We also discuss the commuting problem: if $g$ is a non-constant bounded holomorphic function on $ D^n$, then what conditions does a bounded function $f$ need to satisfy so that $T_f$ commutes with $T_g$?

For any rotation-invariant positive regular Borel measure $\nu$ on the closed unit ball $\overline{{B}}_n$ whose support contains the unit sphere ${S}_n$, let $L^2_{a}$ be the closure in $L^2 = L^2(\overline{{B}}_n,\mathrm{d}\nu)$ of all analytic polynomials. For a bounded Borel function $f$ on $\overline{{B}}_n$, the Toeplitz operator $T_f$ is defined by $T_f (\varphi) = P(f\varphi)$ for $\varphi\in L^{2}_{a}$, where $P$ is the orthogonal projection from $L^2$ onto $L^{2}_{a}$. We show that if $f$ is continuous on $\overline{{B}}_n$, then $T_f$ is compact if and only if $f(z) = 0$ for all $z$ on the unit sphere. This is well known when $L^{2}_{a}$ is replaced by the classical Bergman or Hardy space.

For any $\alpha>-1$, let $A^2_{\alpha}$ be the weighted Bergman space on the unit ball corresponding to the weight $(1-|\mathbf{z}|^2)^{\alpha}$. We show that if all except possibly one of the Toeplitz operators $T_{f_1},\ldots, T_{f_{r}}$ are diagonal with respect to the standard orthonormal basis of $A^2_{\alpha}$ and $T_{f_1}\cdots T_{f_r}$ has finite rank, then one of the functions $f_1,\ldots,f_r$ must be the zero function.

For $\alpha>-1$, let $A^2_{\alpha}$ be the corresponding weighted Bergman space of the unit ball in $\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_{\alpha}$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}\cdots z_{n}^{L_n}$ for strictly positive integers $L_1,\ldots, L_n$, or $g(z)=|z_1|^{s_1}\cdots |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,\ldots, s_n$ and a bounded measurable function $h$ on $[0,1)$.

Let $L_{a}^2$ denote the Bergman space of the open unit ball $\mathbb{B}_n$ in $\mathbb{C}^n$, for $n\geq 1$. The Toeplitz algebra $\mathfrak{T}$ is the $C^{*}$-algebra generated by all Toeplitz operators $T_{f}$ with $f\in L^{\infty}$. It was proved by D. Suarez that for $n=1$, the closed bilateral commutator ideal generated by operators of the form $T_{f}T_{g}-T_{g}T_{f}$, where $f,g\in L^{\infty}$, coincides with $\mathfrak{T}$. With a different approach, we can show that for $n\geq 1$, the closed bilateral ideal generated by operators of the above form, where $f,g$ can be required to be continuous on the open unit ball or supported in a nowhere dense set, is also all of $\mathfrak{T}$.

Let $\mathfrak{T}$ denote the full Toeplitz algebra on the Bergman space of the unit ball $\mathbb{B}_n$. For each subset $G$ of $L^{\infty}$, let $\mathcal{CI}(G)$ denote the closed two-sided ideal of $\mathfrak{T}$ generated by all $T_fT_g-T_gT_f$ with $f,g\in G$. It is known that $\mathcal{CI}(C(\overline{\mathbb{B}}_n))=\mathcal{K}$-the ideal of compact operators and $\mathcal{CI}(C(\mathbb{B}_n)\cap L^{\infty})=\mathfrak{T}$. Despite these extreme cases, there are subsets $G$ of $L^{\infty}$ so that $\mathcal{K}\subset{\mathcal{CI}}(G)\subset\mathfrak{T}$. This paper gives a construction of a class of such subsets.

Using the joint local mean oscillation, Jingbo Xia showed that the essential commutant of $\mathfrak{T}(\mathcal{L})$, where $\mathcal{L}$ is the subalgebra of $L^{\infty}$ generated by all functions which are bounded and have at most one discontinuity, is $\mathfrak{T}(QC)$. Even though Xia's method cannot be used, we are able to generalize his result to Toeplitz operators in higher dimensions with a different approach. This result is stronger than the well-known result stating that the essential commutant of the full Toeplitz algebra $\mathfrak{T}$ is $\mathfrak{T}(QC)$.