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.     

Essential spectrum of difference operators on periodic metric spaces

The paper deals with the study of Fredholm property and essential spectrum of general difference (or band) operators acting on the spaces l ^p (X) on a discrete metric space X periodic with respect to the action of a finitely generated discrete group. The Schrödinger operator on a combinatorial periodic graph is a prominent example of a band operator of this kind.     

Quasi-banded operators,convolutions with almost periodic or quasi-continuous data,and their approximations

We study the stability and Fredholm property of the finite sections of quasi-banded operators acting on L^p$L^p$ spaces over the real line. This family is significantly larger than the set of band-dominated operators, but still permits to derive criteria for the stability and results on the splitting property, as well as an index formula in the form as it is known for the classical cases. In particular, this class covers convolution type operators with semi-almost periodic and quasi-continuous symbols, and operators of multiplication by slowly oscillating, almost periodic or even more general coefficients.     

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR ⁿ , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL ^p (), where is a nonatomic measure (1p<).     

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     

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Fredholm conditions and an index formula are obtained for Wiener-Hopf operators W(a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces where 1 < p < ∞, w is a Muckenhoupt weight on and . The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on that are continuous on and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SO_{p, w} of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calderón-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a. Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.     

Zeros of Fredholm Operator Valued Hp–Functions

This paper is concerned with Fredholm operator valued H^p – functions on the unit disc, where the Fredholm operators action a Banach space. Sufficient conditions are presented which guarantee that Fatou's theorem is valid. Using the theory of traces and determinants on quasi – Banach operator ideals, we develop conditions that guarantee that the zeros of Fredholm operator valued H^p – functions satisfy the Blaschke condition.     

Nonclassical multidimensional singular integral operators on R + n+1

Banach algebras of certain bounded operators acting on the half-spaceL _p (R ₊ ⁿ⁺¹ ,x ₀ ) (1<p<, –1<<p–1) are defined which contain for example Wiener-Hopf operators, defined by multidimensional singular convolution integral operators, as well as certain singular integral operators with fixed singularities. Moreover the symbol may be a positive homogeneous function only piecewise continuous on the unit sphere. Actually these multidimensional singular integral operators may be not Calderón-Zygmund operators but are built up by those in lower dimensions. This paper is a continuation of a joint paper of the author together with R.V. Duduchava [10]. The purpose is to investigate invertibility or Fredholm properties of these operators, while the continuity is given by definition. This is done in [10] forp=2 and –1<<1, and in the present paper forL _p (R ₊ ⁿ⁺¹ ,x ₀ ) with 1<p< and –1<<p–1.     

The C*-algebra of the N-dimensional harmonic oscillator

In this paper we study the Fredholm thoery of a C*-algebraOl of o-order pseudo-differential operators on L²(ⁿ). IfK denotes the ideal of all compact operators of L², the algebraOl will be generated by (i) the idealK, (ii) a function algebra CS(ⁿ) and (iii) by the bounded operators x_j, D_j, j=1,...,n, = H^–1/2, H=1+¦x¦²–. We show thatOl/K is a commutative C*-algebra with identity and obtain its Gelfany space M. This provides Fredholm criterion and index formula for a graded algebra of partial differential operators including all oeprators with polynomial coefficients. We also give Fredholm criterion and index formula for systems of such operators.     

Fredholm and Propernes Properties of Quasilinear Elliptic Operators on RN

We discuss the Fredholm and properness properties of second-order quasilinear elliptic operators viewed as mappings from W^2,p( R ^N) to L^p( R ^N) with N < p < ∞. The unboundedness of the domain makes the standard Sobolev embedding theorems inadequate to investigate such issues. Instead, we develop several new tools and methods to obtain fairly simple necessary and suffcient conditions for such operators to be Fredholm with a given index and to be proper on the closed bounded subsets of W^2,p( R ^N). It is noteworthy that the translation invariance of the domain, well-known to be responsible for the lack of compactness in the Sobolev embedding theorems, is taken advantage of to establish results in the opposite direction and is indeed crucial to the proof of the properness criteria. The limitation to second-order and scalar equations chosen in our exposition is relatively unimportant, as none of the arguments involved here relies upon either of these assumptions. Generalizations to higher order equations or to systems are thus clearly possible with a variableamount of extra work. Various applications, notably but not limited, to global bifurcation problems, are described elsewhere.     

On the Equivalence of Spectra of Linear Partial Differential Operators in Certain Function Spaces

For linear differential operators with coefficients of class C on ⁿ, we prove theorems on the simultaneous invertibility and equivalence of spectra in the Lebesgue space L ^p, Stepanov space M ^p, and in a particular Banach space V ^p L ^p, p 1     

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     

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 < ∞.     

偶数阶自共轭微分算子谱的离散性条件

In this paper, we consider differential operators of 2nd-order
a[u]=(-1)^k(a_k(x)u^(k)(x))(k), x∈(0, ∞)
whose coefficients a_k(x) are restricted by powers of e^x, and give conditions on the coefficients sufficient to ensure that the spectrum is discrete; next we formulate necessary and sufficient conditions for the discreteness of the spectrum of differential operators whose coefficients a_k(x) may increase as e^αkx as x→∞.     

The spectrum of the Leray transform for convex Reinhardt domains in

The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in . Our class is self-dual; it contains some domains with less than C²-smooth boundary and also some domains with smooth boundary and degenerate Levi form. L²-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.     

Fredholmness of pseudodifference operators in weighted Spaces

The aim of the paper is to present some results concerning pseudodifference operators on ?^N, which are a discrete analog of standard pseudodifferential operators on ?^N. We study the Fredholm property of pseudodifference operators acting in weighted l ^p spaces on ?^N and the Phragmen-Lindelöf principle for solutions of pseudodifference equations and give applications of these results to discrete Schrö dinger operators on ?^N.     

Boundedness and Fredholmness of Pseudodifferential Operators in Variable Exponent Spaces

We prove a statement on the boundedness of a certain class of singular type operators in the weighted spaces with variable exponent p(x) and a power type weight w, from which we derive the boundedness of pseudodifferential operators of H?rmander class S ⁰ _1,0 in such spaces. This gives us a possibility to obtain a necessary and sufficient condition for pseudodifferential operators of the class OPS ^m _1,0 with symbols slowly oscillating at infinity, to be Fredholm within the frameworks of weighted Sobolev spaces with constant smoothness s, variable p(·)-exponent, and exponential weights w. Supported by CONACYT Project No.43432 (Mexico), the Project HAOTA of CEMAT at Instituto Superior Técnico, Lisbon (Portugal) and the INTAS Project “Variable Exponent Analysis” Nr.06-1000017-8792.     

Polynomially Compact-Like Strongly Continuous Semigroups

J. R. Cuthbert gave some results about the class of semigroups of operators (T(t)) _t0 on a Banach space X which have the property that for some t>0, T(t)–I is compact. Cuthbert's results were extended to various classes of operators generalizing the set of compact operators such as the ideal of Fredholm perturbations or the set of Riesz operators. The purpose of the present paper is to give further results in this direction. Thus we consider semigroups for which there exists a non-trivial polynomial p()C[z] such that, for some t>0, p(T(t))J(X) where J(X) is an arbitrary proper two-sided ideal of the algebra (X) contained in the set of Fredholm perturbations.