On the Essential Lower Bound of Elements in von Neumann Algebras

In this paper we obtain a number of interesting results on the connection that exists between the essential lower bound of an element in a von Neumann algebra and its Fredholm properties. These connections lead to an extension of a well-known perturbation theorem on Fredholm operators of index zero. Finally we obtain for special cases necessary and sufficient conditions for elements in the closure of the group of invertible elements to be essentially topological divisor of zero     

Exponential decay and Fredholm properties in second-order quasilinear elliptic systems

We consider second-order quasilinear elliptic systems on unbounded domains in the setting of Sobolev spaces. We complete our earlier work on the Fredholm and properness properties of the associated differential operators by giving verifiable conditions for the linearization to be Fredholm of index zero. This opens the way to using the degree for C¹-Fredholm maps of index zero as a tool in the study of such quasilinear systems. Our work also enables us to check the Fredholm assumption which plays an important role in Rabier's approach to proving exponential decay to zero at infinity of solutions.     

Normal operators in -algebras without nice approximants

The second author constructed a separable direct limit -algebra with real rank zero containing a normal element whose spectrum is the closed unit disk that is not the limit of normal elements in the limiting algebras, and is not a limit of normals in the algebra having finite spectrum. We use Fredholm index theory to modify and simplify this construction to obtain such examples that are not limits of any ``nice' types of elements.

     

On multidimensional difference operators and equations

We study the solvability of multidimensional difference equations in Sobolev–Slobodetskii spaces. In the simplest model case, we describe the solvability picture for such equations. In the general case, we present conditions for the Fredholm property and a theorem on the zero index.     

Perturbing Fredholm operators to obtain invertible operators

A condition is given on a set $\text{Ol}$ of operators on Hilbert space that guarantees it has the following property: For any Fredholm operator T of index zero there exists anA?$\text{A}$ such that T + ?A is invertible for all sufficiently small nonzero ?. As a corollary one obtains in a quite general setting the density of the invertible Toeplitz operators in the set of Fredholm Toeplitz operators of index zero.     

Convergence of a family of Galerkin discretizations for the Stokes–Darcy coupled problem

In this article we analyze the well‐posedness (unique solvability, stability, and Céa's estimate) of a family of Galerkin schemes for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers—Joseph—Saffman law. We consider the usual primal formulation in the Stokes domain and the dual‐mixed one in the Darcy region, which yields a compact perturbation of an invertible mapping as the resulting operator equation. We then apply a classical result on projection methods for Fredholm operators of index zero to show that use of any pair of stable Stokes and Darcy elements implies the well‐posedness of the corresponding Stokes—Darcy Galerkin scheme. This extends previous results showing well‐posedness only for Bernardi—Raugel and Raviart—Thomas elements. In addition, we show that under somewhat more demanding hypotheses, an alternative approach that makes no use of compactness arguments can also be applied. Finally, we provide several numerical results illustrating the good performance of the Galerkin method for different geometries of the problem using the MINI element and the Raviart—Thomas subspace of lowest order. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 721–748, 2011     

An Invertibility Criterion in a C*-Algebra Acting on the Hardy Space with Applications to Composition Operators

In this paper, we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators, we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application, we prove that for “quasi-parabolic” composition operators the spectra and the essential spectra are equal.     

The Equivalence between Property (ω) and Weyl’s Theorem

We call T ∈ B(H) consistent in Fredholm and index (briefly a CFI operator) if for each B ∈ B(H),T B and BT are Fredholm together and the same index of B,or not Fredholm together.Using a new spectrum defined in view of the CFI operator,we give the equivalence of Weyl’s theorem and property (ω) for T and its conjugate operator T* .In addition,the property (ω) for operator matrices is considered.     

On the fredholm theory of a planar problem with shift for a pair of functions

We establish necessary and sufficient conditions for the Fredholm property of a planar problem with shift and conjugation for a pair of functions and obtain a formula for the determination of its index.     

Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group

In the paper Wiener-Hopf operators on a semigroup of nonnegative elements of a linearly quasi-ordered torsion free Abelian group are considered. Wiener-Hopf factorization of an invertible element of the group algebra is constructed, notions of a topological index and a factor index are introduced. It turns out that the set of factor indices for invertible elements of the group algebra is a linearly ordered group. It is shown that Wiener-Hopf operator with an invertible symbol is an one-side invertible operator and its invertibility properties are defined by the sign of the factor index of its symbol. Groups on which there exist nontrivial Fredholm Wiener-Hopf operators are described. As an example, all linear quasi-orders on the group ⁿ are found and corresponding Wiener-Hopf operators are considered.     

Wely定理与$(\omega)$性质的等价性

We call T C B（H） consistent in Fredholm and index （briefly a CFI operator） if for each B ∈ B（H）, TB and BT are Fredholm together and the same index of B, or not Fredholm together. Using a new spectrum defined in view of the CFI operator, we give the equivalence of Weyl＇s theorem and property （ω） for T and its conjugate operator T^＊. In addition, the property （ω） for operator matrices is considered.     

The Fredholm Index for Elements of Toeplitz-Composition C<Superscript>*</Superscript>-Algebras

We analyze the essential sectrum and index theory of elements of Toeplitz-composition C^*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C^*-algebra for the action of the symbol on the boundary circle. Using this result we obtain sufficient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse.     

Toeplitz algebra and Hankel algebra on the harmonic Bergman space

In this paper we completely characterize compact Toeplitz operators on the harmonic Bergman space. By using this result we establish the short exact sequences associated with the Toeplitz algebra and the Hankel algebra. We show that the Fredholm index of each Fredholm operator in the Toeplitz algebra or the Hankel algebra is zero.     

Elliptic translators on manifolds with multidimensional singularities

We consider translators on manifolds with many-dimensional singularities. We state the definition of ellipticity for translators, prove a finiteness (Fredholm property) theorem, and establish an index formula.     

A Duduchava-Saginashvili's type theory for Wiener-Hopf plus Hankel operators

This paper presents a Duduchava-Saginashvili's type theory for Wiener-Hopf plus Hankel operators with semi-almost periodic Fourier symbols and acting between Lp Lebesgue spaces. This means the obtainment of one-sided invertibility and Fredholm property for these operators upon certain mean values of the representatives at infinity of their Fourier symbols. Additionally, a formula for the Fredholm index is provided by introducing a corresponding winding number of some new elements.     

The Euler characteristic is stable under compact perturbations

We prove in the general case the stability under compact perturbations of the index (i.e. the Euler characteristic) of a Fredholm complex of Banach spaces. In particular, we obtain the corresponding stability property for Fredholm multioperators. These results are the consequence of a similar statement, concerning more general objects called Fredholm pairs.

     

Helmholtz equation outside an open arc in a plane with a mixed boundary condition

A boundary value problem for the Helmholtz equation outside an open arc in a plane is studied with mixed boundary conditions. In doing so, the Dirichlet condition is specified on one side of the open arc and the boundary condition of the third kind is specified on the other side of the open arc. We consider non-propagative Helmholtz equation, real-valued solutions of which satisfy maximum principle. By using the potential theory the boundary value problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original boundary value problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hardy空间H~1(S)上Toeplitz算子的Fredholm性质

本文讨论了Toeplitz算子在Hardy空间H1(S)上的Fredholm性质.证明了在单位球面S上处处不为零的连续函数φ具有对数有界平均振动时,以其为符号的Toeplitz算子Tφ是Fredholm算子,并且此时Tφ的指标为零.     

On the Invertible Boundary Integral Operator in the Exterior Impedance Problem for the Helmholtz Equation

We propose an approach for reduction of the impedance problem for propagative Helmholtz equation outside several obstacles to the uniquely solvable Fredholm integral equation of the second kind and index zero. The integral equation in this approach is derived by introducing auxiliary boundaries with an appropriate boundary conditions inside obstacles.     

一致Fredholm指标算子及广义(ω′)性质

本文给出了一致Fredholm指标算子的定义及判定,同时定义了Weyl型定理的一种新变化:广义(ω')性质.根据一致Fredholm指标性质定义出一种新的谱集,通过该谱集给出了Hilbert空间上有界线性算子满足广义(ω')性质的充要条件,并且研究了广义(ω')性质的摄动,还研究了算子的亚循环性和广义(ω')性质之间的关系.