BCL index and Fredholm tuples

In this paper we will prove that the BCL index for -algebras generated by two essentially doubly commuting isometries is equal to the index of the Fredholm tuples formed by these two isometries. We will then compute this index for certain sub-Hardy modules over the bidisk. Some interesting corollaries are also listed.

     

On Fredholm index in Banach spaces

We prove the stability of the index of a Fredholm complex of Banach spaces under those compact perturbations which are uniform limits of finite-rank operators (Theorem 3.3). This result is a consequence of some similar statements (Theorems 3.1 and 3.2) concerning more general objects, namely the Fredholm pairs (Definition 1.1).     

On the index of Fredholm pairs of idempotents

Using the technique of block-operators, in this note, we prove that if P and Q are idempotents and （P - Q）^2n＋1 is in the trace class, then （P - Q）^2m＋1 is also in the trace class and tr（P - Q）^2m＋1 = dim（k（P） ∩ k（Q）^⊥） -dim（k（P）^⊥ N k（Q））, for all m ≥ n. Moreover, we prove that dim（k（P）∩ k（Q）^⊥） = dim（k（P）^⊥ ∩k（Q）） if and only if there exists a unitary U such that UP = QU and PU = UQ, where k（T） denotes the range of T. Keywords Fredholm, orthogonal projection, positive operator     

Quotient morphisms, compositions, and Fredholm index

In this paper we define a composition between two quotient morphisms and prove a multiplication formula for the index of the composition of two Fredholm quotient morphisms. Using this formula and the fact that any linear relation can be seen as a quotient morphism, we obtain a multiplication formula for the index of the composition of two Fredholm linear relations.     

Fredholm index and spectral flow in non-self-adjoint case

A version of the "Fredholm index = spectral flow" theorem is proved for general families of elliptic operators {A(t)} t∈R on closed (compact and without boundary) manifolds. Here we do not require that A(t), t∈R or its leading part is self-adjoint.     

Computing the Fredholm index of Toeplitz operators with continuous symbols

We show how to compute the Fredholm index of a Toeplitz operator with a continuous symbol constructed from any subnormal operator with compact self-commutator. We also show that the essential spectral pictures of such Toeplitz operators can be prescribed arbitrarily.

     

Degree theory forC 1 Fredholm mappings of index 0

We develop a degree theory forC ¹ Fredholm mappings of index 0 between Banach spaces and Banach manifolds. As in earlier work devoted to theC ² case, our approach is based upon the concept of parity of a curve of linear Fredholm operators of index 0. This avoids considerations about Fredholm structures involved in other approaches and leads to a theory as complete as that of Leray-Schauder in a much broader setting. In particular, the well-known possible sign change under homotopy is fully elucidated. The technical difficulty arising withC ¹ versusC ² Fredholm mappings of index 0 is notorious: with onlyC ¹ smoothness, the Sard-Smale theorem is no longer available to handle crucial issues involving homotopy. In this work, this difficulty is overcome by using a new approximation theorem forC ¹ Fredholm mappings of arbitrary index instead of the Sard—Smale theorem when dealing with homotopies.     

The Fredholm index of a pair of commuting operators

This paper concerns Fredholm theory in several variables, and its applications to Hilbert spaces of analytic functions. One feature is the introduction of ideas from commutative algebra to operator theory. Specifically, we introduce a method to calculate the Fredholm index of a pair of commuting operators. To achieve this, we define and study the Hilbert space analogs of Samuel multiplicities in commutative algebra. Then the theory is applied to the symmetric Fock space. In particular, our results imply a satisfactory answer to Arveson’s program on developing a Fredholm theory for pure d-contractions when d = 2, including both the Fredholmness problem and the calculation of indices. We also show that Arveson’s curvature invariant is in fact always equal to the Samuel multiplicity for an arbitrary pure d-contraction with finite defect rank. It follows that the curvature is a similarity invariant. Received: October 2004 Revision: May 2005 Accepted: May 2005 Partially supported by National Science Foundation Grant DMS 0400509.     

On stability of index of Fredholm complexes on the C* -algebra

This article deals with the index of Fredholm complexes of Λ-operators of the Hilbert Λ-modulus on C*-algebra. For this class of operators necessary and sufficient conditions in order to be a Fredholm, are obtained. Based on these results, a notion of Fredholm complex and its index is introduced. For this index, a stability theorem related to various perturbations is proved. In the second part of the article, a completation of a semigroup Fredholm complexes is analysed. It is proved that the group K _G (X, Λ) is the completation of G ? Λ-fibration of the above group on the compact space X.     

Multiple limit point bifurcation for Fredholm operators of non-zero index

Summary Using the method of embedding a given equation into a higher dimensional problem, examples of multiple limit point solutions emanating from a single limit point are constructed within the context of algebraic and ordinary differential equations. In both instances, the linearised problem around the limit point is assumed to lead to non-self-adjoint operators.

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Résumé En utilisant la méthode d'inclusion d'une équation donnée dans un problème de dimension supérieure, on construit à partir d'un point critique simple des exemples de solutions ayant un point critique multiple, dans le contexte des équations algébriques et des équations différentielles ordinaires. Dans les deux cas, on suppose que le problème linéarisé autour du point critique engendre des opérateurs non-autoadjoints.

     

A global bifurcation index for set-valued perturbations of Fredholm operators

In the paper the homotopy invariant detecting global bifurcations of solutions to multi-parameter equations involving general set-valued perturbations of linear Fredholm operators of non-negative index is introduced. Some applications to the existence problems for differential inclusions are provided.     

The Fredholm index of a pair of commuting operators, II

We first show that an inequality on Hilbert modules, obtained by Douglas and Yan in 1993, is always an equality. This allows us to establish the semi-continuity of the generalized Samuel multiplicities for a pair of commuting operators. Then we discuss the general structure of a Fredholm pair, aiming at developing a model theory. For application we prove that the Samuel additivity formula on Hilbert spaces of holomorphic functions is equivalent to a generalized Gleason problem. As a consequence it follows the additivity of Samuel multiplicity, in its full generality, on the symmetric Fock space. During the course we discover that a variant e^′(⋅) of the classic algebraic Samuel multiplicity might be more suitable for Hilbert modules and can lead to better results.     

On the Fredholm alternative

We give a proof of the Fredholm alternative for compact operators being adjoint with respect to a nondegenerate bilinear form which is not necessarily bounded.     

The index for Fredholm elements in a Banach algebra via a trace II

We show that the index defined via a trace for Fredholm elements in a Banach algebra has the property that an index zero Fredholm element can be decomposed as the sum of an invertible element and an element in the socle. We identify the set of index zero Fredholm elements as an upper semiregularity with the Jacobson property. The Weyl spectrum is then characterized in terms of the index.     

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.      

Fredholm determinants

The article provides with a down to earth exposition of the Fredholm theory with applications to Brownian motion and KdV equation.