Toeplitz and composition operators on H2 (B n)

The composition operators with closed rangc on H²(B _n) are characterized, and the Fredholmness of products of Toeplitz and composition operators discussed. Moreover, using composition operators, the spectra of Toeplitz operators are studied. Project supported by the National Natural Sciencc Foundation of China and the Postdoctoral Science Foundation of China.     

Some properties of composition operators on Hilbert spaces of Dirichlet series

In this paper, we study some properties of composition operators on Hilbert spaces of Dirichlet series, which include the Fredholmness, Hilbert-Schmidtness, spectra, cyclic and hypercyclic phenomenons, and also answer a norm question raised by Cowen and MacCluer.     

Weighted composition operators on the Hilbert space of Dirichlet series

In this paper, we study weighted composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The Hermitianness, Fredholmness and invertibility of such operators are characterized, and the spectra of compact and invertible weighted composition operators are also described.     

Fredholmness of singular integral operators with discrete subexponentialgroups of shifts on Lebesgue spaces

The paper is devoted to an application of a general local method of studying the Fredholmness of nonlocal bounded linear operators to Banach algebras of singular integral operators with piecewise continuous coefficients and discrete subexponential groups of piecewise smooth shifts acting topologically freely on composed contours. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fredholmness of multipliers on Hardy–Sobolev spaces

This paper discusses the Fredholmness of multipliers on Hardy–Sobolev Spaces and obtains an index formula for the multipliers with some special symbols. Our results show that Hardy–Sobolev spaces have richer properties than classical holomorphic function spaces, and the behavior of the operators on these spaces is complex. Some methods of Hardy or Bergman spaces fail in the case of the Hardy–Sobolev space.     

解析Sobolev型空间上的算子与算子代数

介绍Hardy-Sobolev空间和Fock空间及其算子与算子代数研究方面所做的工作,包括对这两类空间上几类特殊算子有界性、紧性、Fredholm性、指标理论、谱和本性谱、范数和本性范数、Schatten-p类的讨论,以及由它们所生成的C~*-代数的研究.     

On a Boundary Value Problem for Pairs of Polyanalytic Functions

The necessary and sufficient conditions of Fredholmness and the index formula for a boundary value problems for a pair of polyanalytic functions, containing differential operators as well as operators with Carleman shift and conjugations, are obtained. Submitted: May 16, 2007. Accepted: November 5, 2007.     

Fredholmness of Toeplitz operators and corona problems

A meromorphic analogue to the corona problem is formulated and studied and its solutions are characterized as being left-invertible in a space of meromorphic functions. The Fredholmness of Toeplitz operators with symbol G∈(L_∞(R))^2×2 is shown to be equivalent to that of a Toeplitz operator with scalar symbol , provided that the Riemann-Hilbert problem admits a solution such that the meromorphic corona problems with data are solvable. The Fredholm properties are characterized in terms of and the corresponding meromorphic left-inverses. Partial index estimates for the symbols and Fredholmness criteria are established for several classes of Toeplitz operators.     

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on ?^∞(Z,U), with U being a complex Banach space, the injectivity of all limit operators of A already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of A, which, on the other hand, is often equivalent to the Fredholmness of A. As a consequence, for operators A in the Wiener algebra, we can characterize the essential spectrum of A on ?p(Z,U), regardless of p∈[1,∞], as the union of point spectra of its limit operators considered as acting on ?^∞(Z,U).     

Toeplitz operators with frequency modulated semi-almost periodic symbols

It is well known that amplitude modulation does not affect Fredholmness of Toeplitz operators. The same is true for frequency modulation provided the symbol of the operator is piecewise continuous. In this article, it is shown that frequency modulation can destroy Fredholmness for Toeplitz operators with almost periodic symbols; the corresponding example is based on the observation that certain almost periodic functions become semi-almost periodic functions after appropriate frequency modulation. Moreover, this article contains several results that can be employed in order to decide whether a Toeplitz operator with a frequency modulated semi-almost periodic symbol is Fredholm.     

The Fredholm index of a second order elliptic system on an infinite cylinder

This paper investigates the Fredholmness of a second order elliptic system on an infinite cylinder in various weighted or non-weighted Sobolev spaces, assuming homogeneous Dirichlet boundary conditions and suitable asymptotic behavior of the coefficients. It is shown that the index coincides with the spectral flow of an associated parametrized family of elliptic systems on the cross-section of the cylinder. Dependence or independence of the index upon the functional setting is discussed along with their impact on spectral invariance and elliptic regularity.      

Singular integral operators on weighted variable exponent Lebesgue spaces on composed Carleson curves

The paper is devoted to singular integral operators acting on weighted variable exponent Lebesgue spaces on certain composed Carleson curves. Necessary and sufficient conditions for Fredholmness and an index formula are obtained.     

Toeplitz Products on Bergman–Sobolev Spaces over the Unit Polydisk

In this paper, we characterize the compactness and Fredholmness of Toeplitz operators and Toeplitz products on Bergman–Sobolev spaces over the unit polydisk. We also calculate the essential norm of finite sums of finite Toeplitz products on these spaces.     

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     

Some continuous index theorems

In this paper we give criteria of Fredholmness and index theorems for Toeplitz operators whose symbols take values in a W^*-algebra factor of type II₁.     

解析函数的Banach空间上之复合算子

本文研究了一类解析函数的Ｂａｎａｃｈ空间Ｘ上之复合算子，这类空间包含了Ｂｌｏｃｈ空间，并且可看作Ｂｅｒｇｍａｎ空间Ｌ１ａ（Ｄ）中具有原子分解的解析函数的对偶空间．我们刻划了这类空间上紧复合算子及Ｆｒｅｄｈｏｌｍ复合算子的特征，此外，还研究了具有闭值域的复合算子．     

一类解析算子函数空间上的复合算子

本文用算子函数论的方法，研究了解析算子函数的Banach空间X，X0上的复合算子．给出此复合算子为有界的条件，并刻划了此复合算子在X0上为紧的特征．     

Pseudodifferential operator method in a problem on the diffraction of an electromagnetic wave on a dielectric body

We study the problem on the diffraction of electromagnetic waves on a solid body in free space. To analyze the integro-differential equations describing this phenomenon, we use the theory of pseudodifferential operators. We evaluate the asymptotic expansion of the symbol and prove the ellipticity and Fredholmness with index zero of the problem operator.     

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.     

Lambert multipliers between <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

In this paper Lambert multipliers acting between L ^p spaces are characterized by using some properties of conditional expectation operator. Also, Fredholmness of corresponding bounded operators is investigated.