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The spectrum of a weighted composition operator $$C_{\psi , \varphi }$$ that is induced by an automorphism has been investigated for over 50 years. However, many results are got only under the condition that the weight function $$\psi $$ is continuous up to the boundary. In this paper we study the spectra and essential spectra of $$C_{\psi ,\varphi }$$ on weighted Bergman spaces when $$\varphi $$ is an elliptic automorphism, without the assumption that $$\psi $$ is continuous up to the boundary.

Analyzing large prehistoric rock avalanches provides significant data for evaluating the disaster posed by these relatively infrequent but destructive geological events. This paper attempts to study the characteristics and dynamics of the Ganqiuchi granitic rock avalanche, in the middle of the northern margin of Qinling Mountains, 30 km to the south of Xi’an, Shaanxi Province, China. In plane view, this rock avalanche is characterized by source area, accumulation area and dammed lake area. Based on previous studies, historical records and regional geological data, the major trigger of the Ganqiuchi rock avalanche is considered to be a strong paleo-earthquake with tremendous energy. The in situ deposit block size distributions of the intact rock mass and the debris deposits are presented and analyzed by using a simple model for estimating the number of fragmentation cycles that the blocks underwent. The results show that the primary controlling factor of the fragmentation process is the pre-existing fractures, and there is a relationship between the potential energy and the fragmentation energy: the latter is approximately 20% of the former. Based on the dynamic discrete element technique, the study proposes a four-stage model for the dynamic course of the Ganqiuchi rock avalanche: (1) failing; (2) highspeed sliding; (3) collision with obstacles; (4) decelerated sliding, which has implication for hazard assessment of the potential rock avalanches in China and other countries with similar geological setting.

In this article, we study complex symmetric Toeplitz operators on the Bergman space and the pluriharmonic Bergman space in several variables. Surprisingly, the necessary and sufficient conditions for Toeplitz operators to be complex symmetric on these two spaces with certain conjugations are just the same. Also, some interesting symmetry properties of complex symmetric Toeplitz operators are obtained.

In this paper, we study the atomic decomposition for variable exponent Bergman spaces. We also give some characterizations for the boundedness and compactness of composition operators on these spaces, which corrects a result by Morovatpoor and Abkar in (Mediterr. J. Math. 17:9, 2020). Furthermore, we investigate the difference of two composition operators on variable exponent Bergman spaces.

In this paper, we show that for $$T\in B({\mathcal {H}})$$ , if $${\mathcal {M}}$$ is almost-invariant for T, then every maximal almost-invariant subspace of $${\mathcal {M}}$$ is of codimension 1 in $${\mathcal {M}}$$ , where $${\mathcal {H}}$$ is a separable, infinite-dimensional Hilbert space. We also describe the maximal hyperinvariant subspaces for normal operators with all the dimensions of eigenspaces at most 1 acting on $${\mathcal {H}}$$ . Our result is that for each hyperinvariant subspace, all its maximal hyperinvariant subspaces are also of codimension 1 in it.

In this paper, we present the new characterizations, in terms of the sequence $$\{z^j\}_{j=1}^\infty $$, about the boundedness and essential norm estimation for the products of differentiation and composition operators from logarithmic Bloch spaces to $$\mu $$-Bloch spaces on the unit disk. Then our main results are applied to show the boundedness of a non-trivial product operator $$C_\varphi D^m$$ with $$\varphi (z)=z^2$$.

Let $$S^p$$ be the space of holomorphic functions whose derivative lies in the classical Hardy space $$H^p$$ over the unit disk. We prove in this paper that the composition operator $$C_\varphi $$ with $$\varphi $$ an automorphism is hypercyclic on $$S^p$$ , $$0

In this article, we characterize the disjoint hypercyclicity of finite weighted pseudo-shifts on an arbitrary Banach sequence space. Moreover, we obtain some interesting consequences of this characterization.

In this paper, we characterize disjoint hypercyclic powers of weighted pseudo-shifts on an arbitrary F-sequence space. As a special case, we deduce the equivalent conditions for the disjoint hypercyclicity of finitely many different powers of weighted shifts on \(\ell ^2({\mathbb {Z},\mathcal {K}})\) with weight sequence \(\{A_n\}_{n=-\infty }^{\infty }\) of positive invertible diagonal operators on \(\mathcal {K}\).

In this paper, we completely characterize the finite rank commutator and semicommutator of two monomial-type Toeplitz operators on the weighted Bergman space or on the weighted pluriharmonic Bergman space of the unit polydisk. As a consequence, some interesting new phenomenons appear in operator theory on the unit polydisk which are different from that on the unit disk and the unit ball.

We give some simple norm estimates of the difference of weighted composition operators acting from Lipschitz space $$\mathcal {L}_\alpha $$ to the space $$H^\infty $$ of all analytic functions on the unit disk, and showing that any two bounded weighted composition operators are pathwise connected. We also obtain some necessary or sufficient conditions for the difference to be compact. Unlike the equivalence of the boundedness and compactness for weighted composition operators from $$\mathcal {L}_\alpha $$ to $$H^\infty ,$$ we show that there exists a difference which is bounded, but not compact.

We characterize the boundedness and compactness of the differences of weighted differentiation composition operators $$ {D}_{\varphi_1,{u}_1}^n-{D}_{\varphi_2,{u}_2}^n, $$ where n ∈ ℕ0, u1,u2 ∈ H( $$ \mathbb{D} $$ ), and $$ \varphi $$ 1, $$ \varphi $$ 2 2 S( $$ \mathbb{D} $$ ), from mixed-norm spaces H(p, q, ϕ), where 0 < p, q < ∞ and ϕ is normal, to weighted-type spaces H ∞ .

In this article, we investigate the mean ergodicity of composition operators acting on the Hardy space $$H^p({\mathbb {D}})$$ , $$1\le p

Let $$\varphi $$ be a complex-valued function in the plane $${\mathbb {C}}.$$ The superposition operator is defined by $$ S_\varphi (f)=\varphi \circ f$$ . In this paper, we characterize the nonlinear superposition operators $$S_\varphi $$ acting between the Zygmund-type and Bloch-type spaces in terms of the order and type or the degree of $$\varphi $$ .

The authors characterize the order boundedness of weighted composition operators acting between Dirichlet type spaces.

In this paper we show that the property of disjoint supercyclic operators T1, ⋯, TN satisfying d-supercyclicity criterion on the same Hilbert space is equivalent to disjointness in supercyclicity of the corresponding left multiplication operators induced by T1, ⋯, TN in the strong operator topology. Besides, by the similar discussion, we also obtain that d-hypercyclicity criterion for any T1, ⋯, TN on the same Hilbert space is equivalent to d-hypercyclicity of the corresponding left multiplication operators induced by T1, ⋯, TN in the ‖.‖2 topology.

In this paper, analogues of the Berkson–Porta formula for the infinitesimal generators of one-parameter semigroup of holomorphic self-maps on the polydisk are obtained. We give a necessary and sufficient condition for a holomorphic vector field to be an infinitesimal generator which improves the theorem given by Contreras, de Fabritiis and Diaz-Madrigal.

A bounded linear operator $T$ on a Banach space $X$ is called subspace-hypercyclic for a nonzero subspace $M$ if $orb\left( {T,x} \right) \cap M$ is dense in $M$ for a vector $x \in X$, where $orb (T,x)=\{T^nx: n=0,1,2,\cdots\}$. Similarly, the bounded linear operator $T$ on a Banach space $X$ is called subspace-supercyclic for a nonzero subspace $M$ if there exists a vector whose projective orbit intersects the subspace $M$ in a relatively dense set. In this paper we provide a Subspace-Supercyclicity Criterion and offer two equivalent conditions of this criterion. At the same time, we also characterize other properties of subspace-supercyclic operators.

In this paper we give a Carleson measure characterization for the compact composition operators between Dirichlet type spaces. We use this characterization to show that every compact composition operator on Dirichlet type spaces is compact on the Bloch space.

This paper concerns affine infinitesimal generators of continuous semigroups of holomorphic functions on the polydisk. An easy-to-check necessary and sufficiently condition for an affine vector field to be an infinitesimal generator is given.

In this paper, we give a new characterization for the boundedness of the product of differentiation and composition operators \({C_\varphi D^m}\) acting on Bloch-type spaces and obtain an estimate for its essential norm in terms of the sequence \({\{z^n\}^{\infty}_{n=1}}\), from which the sufficient and necessary condition of compactness of the operator \({C_\varphi D^m}\) follows immediately.

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.

In 2010, Ruhan Zhao obtained the essential norms of composition operators between Bloch type spaces in the disk by the nth power of the induced analytic function. This paper will generalize Zhao’s results to the polydisk. Unlike the case of the composition operators on the unit disk, the essential norms are different for the cases \({p \in (0,1)}\) and p ≥ 1.

In this paper, we characterize the boundedness and compactness of the weighted composition operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the unit ball and estimate the essential norms of the weighted composition operators.

In this paper, we obtain a symmetry number for the commutator of quasi- homogeneous Toeplitz operators on the harmonic Bergman space. Then we use it to characterize the commuting Toeplitz operators with quasihomogeneous symbols. Also, we show that a Toeplitz operator with an analytic or co-analytic monomial symbol commutes with another Toeplitz operator only in the trivial case.

In this paper, we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators, an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit sphere. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L1(Bn) symbol.

We investigate the adjoints of linear fractional composition operators C ϕ acting on classical Dirichlet space D(B N ) in the unit ball B N of ℂ N , and characterize the normality and essential normality of C ϕ on D(B N ) and the Dirichlet space modulo constant function D 0(B N ), where ϕ is a linear fractional map ϕ of B N . In addition, we also show that for any non-elliptic linear fractional map ϕ of B N , the composition maps σ o ϕ and ϕ o σ are elliptic or parabolic linear fractional maps of B N .

In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in \({\mathbb{C}}^{n}\). We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form \(\xi^{k} \varphi\) is studied, where \(k \in {\mathbb{Z}}^{n}\) and φ is a radial function.

In this paper we first look upon some known results on the composition operator as bounded or compact on the Bloch-type space in polydisk and ball, and then give a sufficient and necessary condition for the composition operator to be compact on the Bloch space in a bounded symmetric domain.