Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.     

Commutativity of kth‐order slant Toeplitz operators

In this paper, commutativity of k^th‐order slant Toeplitz operators are discussed. We show that commutativity and essential commutativity of two slant Toeplitz operators are the same. Also, we study k^th‐order slant Toeplitz operators on the Bergman space L²_a(D) and give some commuting properties, algebraic and spectral properties of k^th‐order slant Toeplitz operators on the Bergman space (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unitary equivalence of one-parameter groups of Toeplitz and composition operators

The composition operators on H² whose symbols are hyperbolic automorphisms of the unit disk fixing ±1 comprise a one-parameter group and the analytic Toeplitz operators coming from covering maps of annuli centered at the origin whose radii are reciprocals also form a one-parameter group. Using the eigenvectors of the composition operators and of the adjoints of the Toeplitz operators, a direct unitary equivalence is found between the restrictions to zH² of the group of Toeplitz operators and the group of adjoints of these composition operators. On the other hand, it is shown that there is not a unitary equivalence of the groups of Toeplitz operators and the adjoints of the composition operators on the whole of H², but there is a similarity between them.     

On composition operators onH p -spaces in several variables

In this paper, we prove that a composition operator onH ^p (B) is Fredholm if and only if it is invertible if and only if its symbol is an automorphism onB, and give the representation of the spectra of a class of composition operators. In addition, using composition operator, we discuss intertwining Toeplitz operators. Supported by NNSF and PDSF     

Composition operators between area-type Nevanlinna classes

This paper is concerned with the composition operators between the area-type Nevanlinna classes. Some sufficient and necessary conditions are given in terms of the concept of Carleson measure and the standard techniques of Montel Theorem for the composition operator C _φ: N _a ^p → N _a ^q to be bounded or compact, where 1 < p ⩽ q. Moreover, the inducing maps which induce invertible or Fredholm composition operators on N _a ^p are characterized. __________ Translated from Journal of Wuhan University (Natural Science Edition), 2004, 50(1): 1–5. This work was supported by the National Natural Science Foundation of China under grant number 19771063     

On self-adjointness of the product of two second-order differential operators

In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the productL ₂ L ₁ of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators. Supported by the Royal Society and the National Natural Science Foundation of China and the Regional Science Foundation of Inner Mongolia     

Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for SφSφψ= Sφψ.     

Every composition operator is (mean) asymptotically Toeplitz

Nazarov and Shapiro recently showed that, while composition operators on the Hardy space H² can only trivially be Toeplitz, or even “Toeplitz plus compact,” it is an interesting problem to determine which of them can be “asymptotically Toeplitz.” I show here that if “asymptotically” is interpreted in, for example, the Cesàro (C,α) sense (α>0), then every composition operator on H² becomes asymptotically Toeplitz.     

具有粗糙核的多线性算子在Herz型空间的有界性

§ 1　Introduction and main resultsLet b∈BMO(Rn) and T be a standard Calderon-Zygmund singular integral operator.Define the commutator[b,T] as follows.[b,T] f(x) =b(x) Tf(x) -T(bf) (x) .In [3 ] ,the boundedness ofthe commutator[b,T] wasestablished on Lp(Rn) .There are thesimilar results in [1 ,2 ] when the commutator was substituted with the multilinearoperators generated by the singular integral operator T and a Taylor series A(see thedefinition below) .Recently,many mathematicians h…     

Ergodic properties of operators in some semi-Hilbertian spaces

This article deals with linear operators T on a complex Hilbert space ?, which are bounded with respect to the seminorm induced by a positive operator A on ?. The A-adjoint and A ^1/2-adjoint of T are considered to obtain some ergodic conditions for T with respect to A. These operators are also employed to investigate the class of orthogonally mean ergodic operators as well as that of A-power bounded operators. Some classes of orthogonally mean ergodic or A-ergodic operators, which come from the theory of generalized Toeplitz operators are considered. In particular, we give an example of an A-ergodic operator (with an injective A) which is not Cesàro ergodic, such that T ^?* is not a quasiaffine transform of an orthogonally mean ergodic operator.     

重调和Hardy空间上的Toeplitz 算子

与调和Bergman 空间相对应, 本文研究重调和Hardy 空间h²(D²) 上的Toeplitz 算子. 本文发现, h²(D²) 上的Toeplitz 算子与经典的Hardy 空间、Bergman 空间及调和Bergman 空间上的Toeplitz算子的性质都有很大的差异. 例如解析Toeplitz 算子可以不是半可换及可交换的. 即使半可换, 其中任何一个符号可以不为常数; 即使可交换, 两个符号的非平凡线性组合也不一定是常数. 本文得到了h²(D²) 上两个解析Toeplitz 算子半可换和可换的充分必要条件.     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Spectral theory for algebraic combinations of Toeplitz and composition operators

We determine the essential spectra of algebraic combinations of Toeplitz operators with continuous symbol and composition operators induced by a class of linear-fractional non-automorphisms of the unit disk. The operators in question act on the Hardy space H² on the unit disk. Our method is to realize the C^*-algebra that they generate as an extension of the compact operators by a concrete C^*-algebra whose invertible elements are easily characterized.     

Numerical Ranges of Radial Toeplitz Operators on Bergman Space

A Toeplitz operator T_fT_\phi with symbol f\phi in L^￥(\mathbbD)L^{\infty}({\mathbb{D}}) on the Bergman space A²(\mathbbD)A^{2}({\mathbb{D}}), where \mathbbD\mathbb{D} denotes the open unit disc, is radial if f(z) = f(|z|)\phi(z) = \phi(|z|) a.e. on \mathbbD\mathbb{D}. In this paper, we consider the numerical ranges of such operators. It is shown that all finite line segments, convex hulls of analytic images of \mathbbD\mathbb{D} and closed convex polygonal regions in the plane are the numerical ranges of radial Toeplitz operators. On the other hand, Toeplitz operators T_fT_\phi with f\phi harmonic on \mathbbD\mathbb{D} and continuous on [`(\mathbbD)]{\overline{\mathbb{D}}} and radial Toeplitz operators are convexoid, but certain compact quasinilpotent Toeplitz operators are not.     

Toeplitz and Hankel type operators on the upper half-plane

An orthogonal decomposition of admissible wavelets is constructed via the Laguerre polynomials, it turns to give a complete decomposition of the space of square integrable functions on the upper half-plane with the measurey dxdy. The first subspace is just the weighted Bergman (or Dzhrbashyan) space. Three types of Ha-plitz operators are defined, they are the generalization of classical Toeplitz, small and big Hankel operators respectively. Their boundedness, compactness and Schatten-von Neumann properties are studied.Research was supported by the National Natural Science Foundation of China.     

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

Hankel-Toeplitz type operators onL a 1 (Ω)

This paper studies Hankel and Toeplitz operators on the Bergman spaceL _a ¹ () of bounded symmetric domains. These operators are defined in terms of a certain bounded projection onL ¹(,dV). The main results of the paper include several characterizations for the boundedness and (weak-star) compactness of these Hankel-Toeplitz type operators. When the symbol is conjugate holomorphic, our results here are similar to those obtained by Békollé, Berger, Coburn, and Zhu [2] in theL ²-Bergman space context.Research partially supported by the National Science Foundation     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Weighted Composition Operators of H∞ into α-Bloch Spaces on the Unit Ball

In this paper we characterize the boundedness and compactness of weighted composition operators of H∞ into α-Bloch spaces on the unit ball in C^n.     

Strongly singular convolution operators on the weighted Herz-type Hardy spaces

Let 0<p≤1<q<0, andw ₁ ,w ₂ ∈ A ₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spacesH K^{α, p} _q(w₁; w₂) to the homogeneous weighted Herz spacesK ^{α, p} _q (w₁; w₂), provided α=n(1−1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spacesH K ^{α, p} _q (w ₁;w ₂) is also investigated. Supported by the National Natural Science Foundation of China