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We consider Toeplitz operators Tu with symbol u on the Bergman space of the unit ball, and then study the convergences and summability for the sequences of powers of Toeplitz operators. We first charactreize analytic symbols φ for which the sequence $$T_\varphi ^{*k}$$ f or $$T_\varphi ^k$$ f converges to 0 or ∞ as k → ∞ in norm for every nonzero Bergman function f. Also, we characterize analytic symbols φ for which the norm of such a sequence is summable or not summable. We also study the corresponding problems on an infinite direct sum of Bergman spaces as a generalization of our result.

A famous theorem of Coburn says that a nonzero Toeplitz operator on the Hardy space of the unit disk is injective or its adjoint operator is injective. In this paper we study the corresponding problem on the Hardy space of the bidisk. We first show the Coburn type theorem fails generally on the bidisk. But, we show that certain pluriharmonic symbols or product symbols of one variable functions induce Toeplitz operators satisfying the Coburn type theorem.

Let M be an invariant subspace of $$H^2$$ over the bi-disk and $$N=H^2\ominus M$$ . Let $$S_{z,N},S_{w,N}$$ be the compression of the multiplication operators $$T_z,T_w$$ on $$H^2$$ onto N. For a two-variable inner function $$\theta $$ , let $$M_\theta = \theta M$$ and $$N_\theta =H^2\ominus M_\theta $$ . We shall study the relationship of the ranks of the cross-commutators $$[S_{z,N},S^*_{w,N}]$$ and $$[S_{z,N_\theta },S^*_{w,N_\theta }]$$ . We also characterize M such that rank $$[S_{z,N},S^*_{w,N}]$$ $$\not =$$ rank $$[S_{z,N_\theta },S^*_{w,N_\theta }]$$ for any non-constant inner function $$\theta $$ .

Let M be an invariant subspace of \(H^2\) over the bidisk. Associated with M, we have the fringe operator \(F^M_z\) on \(M\ominus w M\). For \(A\subset H^2\), let [A] denote the smallest invariant subspace containing A. Assume that \(F^M_z\) is Fredholm. If h is a bounded analytic function on \(\mathbb {D}^2\) satisfying \(h(0,0)\not =0\), then \(F^{[h M]}_z\) is Fredholm and \(\mathrm{ind}\,F^{[h M]}_z=\mathrm{ind}\,F^M_z\).

Let ψ δ 1 be the singular inner function associated with the unit point mass at z = 1 . We shall give an infinite dimensional closed subspace M of ker T ψ δ 1 ⁎ , the kernel of the adjoint of the Toeplitz operator T ψ δ 1 , satisfying that for every nonzero v in M, the smallest backward shift invariant subspace of H 2 containing v coincides with ker T ψ δ 1 ⁎ .

We study the kernels of Toeplitz operators on the Hardy space on the bidisk. We first give a sufficient condition for a general symbol to be antiholomorphic under the assumption that the kernel of the corresponding Toeplitz operator contains a backward shift invariant subspace. As an application, we construct an invariant subspace whose its orthogonal complement can not be the kernel of any Toeplitz operator. Also, we give a characterization on a Toeplitz operator for which the orthogonal complement of its kernel is generated by certain inner functions. Finally, we describe all backward shift invariant subspaces which are in the kernels of Toeplitz operators with homogeneous type symbols. As an application, we show that there is a Toeplitz operator for which its kernel has dimension of any given integer and the orthogonal complement of the kernel is generated by two functions. Our result shows that there are higher dimensional phenomena.

Let $H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of $H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with $\overline{\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.

An invariant subspace M of the Hardy space over the bidisk is said to have two side frames if M ⊖ z M and M ⊖ w M contain nonzero T w and T z invariant subspaces, respectively. If one frame is in H 2 ( z ) and the other is in H 2 ( w ) , M is said to have two side rigid frames. We shall show an example of an invariant subspace having two side frames which is not unitarily equivalent to any one having two side rigid frames. We also give some sufficient conditions on M for M to be unitarily equivalent to a rigid one.

We consider Toeplitz operators $$T_\varphi $$ with symbol $$\varphi $$ on the Hardy space $$H^2(\mathbb {D}^n)$$ of the polydisk $$\mathbb {D}^n$$ . We characterize the bounded analytic functions $$\varphi $$ on $$\mathbb {D}^n$$ such that $$\Vert T^{*k}_{\varphi } f\Vert \rightarrow 0$$ as $$k\rightarrow \infty $$ and $$\sum ^\infty _{k=0}\Vert T^{*k}_{\varphi } f\Vert ^2$$ $$

Recently, the study of the structure of closed ideals in $$H^{\infty }$$ whose zero sets are contained in G, the union set of non-trivial Gleason parts, has progressed remarkably. We generalize these results to closed ideals in Douglas algebras A. For non-zero functions $$f_1,f_2,\ldots ,f_n$$ in A, $$I=\sum ^n_{j=1}f_j A$$ is an ideal (may not be closed) in A. We also show that if I is closed in A and its common zero set is contained in G, then $$I=b A$$ for a Carleson–Newman Blaschke product b.

The paper studies the rank of finite Rudin type backward shift invariant subspaces over the bidisk.

We here investigate the topological structure of the space of weighted composition operators boundedly acting between two Hilbert spaces of analytic functions on the open unit disk, satisfying some natural hypotheses. We will consider the operator norm topology and the Hilbert–Schmidt norm topology respectively. These results will be involved in the investigation for the explicit cases of the classical Hardy–Hilbert space, the weighted Bergman spaces and the Dirichlet space. Furthermore we will estimate the Hilbert–Schmidt norms of difference of two composition operators acting from the Dirichlet space to the Hardy and the weighted Bergman spaces.

Let be the family of analytic functions on . Let be an analytic self-map of and for . Let be the weighted composition operator on . It is proved that if contains the disk algebra, then there is with such that is an automorphism of and on .

Kwon and Bae gave an interpolation for a continuous form of Hlder's inequality for a real-valued bounded measurable function on a product of measure spaces. It is given some generalizations of their result.

For a mixed invariant subspace N of H 2 under T z and $T^{*}_{w}$ , in the previous paper we studied the case rank[V z ,V w ]=1 and either dim(N⊖zN)=0 or 1. In this paper, we study the structure of N satisfying rank[V z ,V w ]=1 and dim(N⊖zN)=2. Our study is deeply concerned with the structure of nonextreme points in the closed unit ball of the space of one variable bounded analytic functions.

Properties of the non-analytic sets N(f) associated with functions f ∈ L ∞ are studied. For an F σ-set E in M(L ∞) satisfying m(E) = 0, we show that if N(f) ∩ E = ∅, then . Using this result, we study some relations between non-analytic sets and QC-level sets.

It is studied the linear combinations of composition operators on the Banach space of bounded harmonic functions on the open unit disk. We determine the essential norm of them.

We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and, moreover, be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk.

Let COP =0∩H∞, where0is the little Bloch space on the open unit disk, andA() be the disk algebra on. For non-zero functionsu1,u2,. . .,uN∈A() and distinct analytic self-maps ϕ1,ϕ2,. . .,ϕNsatisfying ϕj∈A() and ∥ϕj∥∞=1 for everyj, it is given characterisations of which the sum of weighted composition operators ∑Nj=1ujCϕjmaps COP intoA().

Let H2 be the Hardy space and H°° the Banach space of bounded analytic functions on the open unit disk ID with the supremum norm || • Hoc. For / £ H°°, we denote by /* the radial limit of / defined a.e. on the boundary 3D of P. We write {|/*| = 1} = {el° £ 3D : \f*(ete)\ — 1}. Let m be the normalized Lebesgue measure on 3D. We denote by 5#* the set of analytic self-maps of D and Sh^.o — G Sh=c : Hvlloo = !}• For p £ Shthe composition operator C^ on H2 is defined by C^f = f o tp for / £ H2. Let C(H2) be the space of composition operators on H2 with the operator norm. The most interesting subject in the study of C(H2) is called the connected component problem, describing the connected component containing a given Cv. It was first studied by Berkson [2]. He showed that if p € 0, then is an isolated point in C(H2). Around 1990, MacCluer [13] and Shapiro and Sundberg [19] revisited the connected component problem on H2. Succeedingly, Bourdon [3], Gallardo-Gutierrez, Gonzalez, Nieminen and Saksman [6] and Moorhouse and Toews [16] followed. Still it remains unclear the connected component problem on H2 (see [4; 17; 18]). Similarly we have the space C(H°°) of composition operators on H°° with the

In the previous paper, we gave a characterization of backward shift invariant subspaces of the Hardy space over the bidisk on which (S z n ; S � ) = 0 for a positive integer n 2. In this case, it holds that S z n = cI for some c 2 C. In this paper, it is proved that if (S φ; S � w ) = 0 and φ 2 H 1 ( z ), then S φ = cI for some c 2 C.

We characterize compactness of sums of weighted composition operators on the Banach algebra of bounded analytic functions on the open unit disk.

We determine the norm and the essential norm of the difference of weighted composition operators on the space of bounded harmonic functions on the open unit disk. The argument is done on the boundary.

Let H∞ be the Banach algebra of bounded analytic functions on the open unit disk D. Let G be the union set of all nontrivial Gleason parts in the maximal ideal space of H∞. Let E be a nonvoid compact and totally disconnected subset of G and nE be a bounded numbering function on E. We characterize nE for which there is a closed ideal I in H∞ such that Z(I)=E and ord(I,x)=nE(x) for every x∈E. Let I1,I2,…,Ik be closed ideals in H∞ satisfying Z(Ii)⊂G for 1⩽i⩽k. We prove that ∏i=1kIi={∏i=1kfi:fi∈Ii,1⩽i⩽k} is a closed ideal. A local ideal theory in H∞ plays an important role to prove our results.

An elementary proof of the Aleman, Richter and Sundberg theorem concerning the invariant subspaces of the Bergman space is given.

Let E be a compact subset of G, the union set of nontrivial Gleason parts, and I(E) be the associate primary ideal of H∞. We give a characterization of the numbering function ord(I(E),x), the zero's order of I(E) at x in E, using the geometrical words of E. We also give some factorization theorems of Blaschke products. Using these, we give some descriptions of the higher order hulls of I(E), and for a Carleson–Newman Blaschke product in I(E) it is proved that I(E) is generated by its subproducts as a closed ideal. When each Ei is ρ-separated, we also prove that the tensor product ⊗i=1kI(Ei) is closed in H∞.

Let \({\mathcal{B}}\) be the Bergman shift on the Bergman space \({L_a^2}\) over the open unit disk and let I be a nontrivial invariant subspace of \({L_a^2}\). Let PI be the orthogonal projection from \({L_a^2}\) onto I. It is proved that \({P_I\mathcal{B}(L_a^2 \ominus I)}\) is not dense in I if and only if \({I \cap \mathcal{D} \ne \{0\}}\), where \({\mathcal{D}}\) is the Dirichlet space. It is also discussed some related topics.

The path connected components are determined in the space of weighted composition operators on the space of bounded harmonic functions with the strong operator topology.

It is known that the structure of invariant subspaces I of the Hardy space H2 over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces \({I\ominus zI}\) completely. In this paper, we study the structure of nontrivial closed subspaces N of H2 with \({T_zN\subset N}\) and \({T^*_wN\subset N}\), which are called mixed invariant subspaces under Tz and \({T^*_w}\). We know that the dimension of \({N\ominus zN}\) ranges from 1 to ∞. If \({T^*_w(N\ominus zN)\subset N\ominus zN}\), we may describe N completely. If \({T^*_w(N\ominus zN)\not\subset N\ominus zN}\), it seems difficult to describe N generally. So we study N under the condition \({dim\,(N\ominus zN)=1}\). Write \({M=H^2\ominus N}\). We describe \({M\ominus wM}\) precisely. We give a characterization of N for which there is a nonzero function \({\varphi}\) in \({M\ominus wM}\) satisfying \({z^k\varphi\in M\ominus wM}\) for every k ≥ 0. We also see that the space \({M\ominus wM}\) has a deep connection with the de Branges–Rovnyak spaces studied by Sarason.

We study composition operators on the space of bounded harmonic functions on the open unit disk. The principal goal of this paper is to provide criteria for determining the essential norm of difference of two composition operators.

This paper demonstrates equivalence amongst the topological structures of path connected components in the spaces of weighted composition operators from $L^{\infty}$, $H^{\infty}$ and the disk algebra into $L^{\infty}$ on the unit circle.

Let $\varphi$ be an analytic self-map of the open unit disk. It is given several sufficient conditions on $\varphi$ for which there is $u\in H^2 \ H^{\infty}$ such that the weighted composition operator $M_u C_\varphi$ on $H^2$ is bounded.

We discuss on invariant subspaces of H 2 ( Γ 2 ) on which rank [ R w , R z ∗ ] = 1 .

Let be the family of inner functions whose non-trivial Frostman shifts are Carleson–Newman Blaschke products. It is proved that each compact subset of the unit circle is the support of a positive singular measure for which the associated singular inner function belongs to .

Gorkin and Mortini introduced the concept of k-hulls, k(x), of points x in M(H ∞) ∖ ∖D, and studied the ideal structures of H ∞ and H ∞ +C. They posed a problem for which x∈ M(H ∞) ∖ ∖D the set I(k(x)) is a closed prime ideal. In this article, we give a partial answer for sparse points x.

We study Hankel-type operators on the space of bounded harmonic functions on the open unit disk. These operators are related to tight uniform algebras, the Dunford-Pettis property, and Bourgain algebras.

We study properties of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology. Moreover, we characterize the compactness of differences of two weighted composition operators.

We characterize polynomials p having leading terms over C2 in a generalized sense. As applications, we study a quasi-invariant subspace [p] generated by p in the Fock space, and show that a quasi-invariant subspace M is similar to [p] if and only if M=[q] for some polynomial q having the same leading term as p. These are generalizations of Guo and Hou's work.

For an invariant subspace I of the Bergman space \(L_a^2 (D)\) on the unit disk D, the associated inner space I ⊖ zI has been known to have nice properties K. Zhu has recently given, in terms of kernels of Hankel operators, several characterizations for an inner space to be maximal. We show that maximality of inner spaces can be understood alternatively by use of the adjoint operator of the Bergman shift operator on \(L_a^2 (D).\)

We answer Mortiniʼs question concerning the level sets on M ( H ∞ ) of a certain harmonic function affirmatively.