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This paper shows some new phenomenon in the spectral theory of Toeplitz operators on the Bergman space, which is considerably different from that of Toeplitz operators on the Hardy space. On the one hand, we prove that the spectrum of the Toeplitz operator with symbol z+p is always connected for every polynomial p with degree less than 3. On the other hand, we show that for each integer k greater than 2, there exists a polynomial p of degree k such that the spectrum of the Toeplitz operator with symbol z+p is a nonempty finite set. Then these results are applied to obtain a new class of non-hyponormal Toeplitz operators with bounded harmonic symbols on the Bergman space for which Weyl's theorem holds. [ABSTRACT FROM AUTHOR]

In this paper, we give a new proof of the Bessonov theorem [Proc. Amer. Math. Soc. 142 (2014), pp. 1301-1313] on finite rank truncated Toeplitz operators on model spaces via the product of Hankel operators on Hardy space. [ABSTRACT FROM AUTHOR]

In this paper, we obtain sufficient and necessary conditions for big truncated Hankel operators on model spaces to be zero, or of finite rank or compact. Our main tools are the properties of Hardy Hankel operators and function algebras. [ABSTRACT FROM AUTHOR]

We give a new and simple compactness criterion for composition operators $C_varphi $ on BMOA and the Bloch space in terms of the norms of $varphi ^n$ in the respective spaces. [ABSTRACT FROM AUTHOR]

AbstractMotivated by recent works of Ahern and$$\breve{\rm c}$$u$$\breve{\rm c}$$kovi? on the disk, we study the generalized zero product problem for Toeplitz operators acting on the Bergman space of the polydisk. First, we extend the results to the polydisk. Next, we study the generalized compact product problem. Our results are new even on the disk. As a consequence on higher dimensional polydisks, we show that the generalized zero and compact product properties are the same for Toeplitz operators in a certain case. [ABSTRACT FROM AUTHOR]

In this paper we use the beautiful formula of Faa di Bruno for the $n$th derivative of composition of two functions to obtain the generalized Schwarz-Pick estimates. By means of those estimates we show that the hyperbolic derivative of an analytic self-map of the unit disk is Lipschitz with respect to the pseudohyperbolic metric. [ABSTRACT FROM AUTHOR]

Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators. [ABSTRACT FROM AUTHOR]

We consider the topological space of all composition operators on the Banach algebra of bounded analytic functions on the unit disk. We obtain a function theoretic characterization of isolated points and show that each isolated composition operator is essentially isolated. [ABSTRACT FROM AUTHOR]

In this note we investigate conditions under which a holomorphic self-map of the unit disk induces a composition operator with closed range on the Bloch space. [ABSTRACT FROM AUTHOR]

By making use of $\mathcal M$-harmonic function theory, we characterize commuting Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball or on the Hardy space of the unit sphere in $n$-dimensional complex space. [ABSTRACT FROM AUTHOR]

In this paper it is shown that two analytic Toeplitz operators essentially doubly commute if and only if they doubly commute on the Bergman space of the polydisk. [ABSTRACT FROM AUTHOR]