Samuel multiplicity and the structure of semi-Fredholm operators

Two numerical invariants refining the Fredholm index are introduced for any semi-Fredholm operator in such a way that their difference calculates the Fredholm index. These two invariants are inspired by Samuel multiplicity in commutative algebra, and can be regarded as the stabilized dimension of the kernel and cokernel. A geometric interpretation of these invariants leads naturally to a 4×4 uptriangular matrix model for any semi-Fredholm operator on a separable Hilbert space. This model can be regarded as a refined, local version of the Apostol's 3×3 triangular representation for arbitrary operators. Some classical results, such as Gohberg's punctured neighborhood theorem, can be read off directly from our matrix model. Banach space operators are also considered.     

MINIMUM MODULI IN VON NEUMANN ALGEBRAS

Abstract

In this paper we answer a question raised in [12] in the affirmative, namely that the essential minimum modulus of an element in a von Neumann algebra, relative to any norm closed two-sided ideal, is equal to the minimum modulus of the element perturbed by an element from the ideal. As a corollary of this result, we extend some basic perturbation results on semi-Fredholm elements to a von Neumann algebra setting. We then characterize the semi-Fredholm elements in terms of the points of continuity of the essential minimum modulus function.     

Samuel multiplicity and the structure of essentially semi-regular operators: A note on a paper of Fang

Motivated by a paper of Fang (2009), we study the Samuel multiplicity and the structure of essentially semi-regular operators on an infinite-dimensional complex Banach space. First, we generalize Fang’s results concerning Samuel multiplicity from semi-Fredholm operators to essentially semi-regular operators by elementary methods in operator theory. Second, we study the structure of essentially semi-regular operators. More precisely, we present a revised version of Fang’s 4 × 4 upper triangular model with a little modification, and prove it in detail after providing numerous preliminary results, some of which are inspired by Fang’s paper. At last, as some applications, we get the structure of semi-Fredholm operators which revised Fang’s 4 × 4 upper triangular model, from a different viewpoint, and characterize a semi-regular point λ ∈ ? in an essentially semi-regular domain.     

Semi-Fredholm operators and sequence conditions

A typical result of this paper is the following: A closed linear operator T between Banach spaces X and Y is lower semi-Fredholm iff it is a rk-(=relatively compact)surjection, i.e. for every bounded sequence (y_n) in Y there is a bounded sequence (x_n) in D(T) such that (y_n?Tx_n) is relatively compact in Y. The concept of c_o-(rk- and wk-)injections and surjections are introduced to characterize semi-Fredholm properties (in the usual or generalized sense). This is done also by new operator moduli.     

Essential spectra through local spectral theory

Based on a nice observation of Eschmeier, this is a study of the use of local spectral theory in investigations of the semi-Fredholm spectrum of a continuous linear operator. We also examine the retention of the semi-Fredholm spectrum under weak intertwining relations; it is shown, inter alias, that if two decomposable operators are intertwined asymptotically by a quasi-affinity then they have identical semi-Fredholm spectra. The results are applied to multipliers on commutative semisimple Banach algebras.

     

半Fredholm算子的一个注记

本文研究了(上,下)半Fredholm算子的结构问题.利用经典的算子理论方法,由闭不变子空间诱导的两个映射,获得了(上,下)半Fredholm算子的多种新的等价刻画,推广了(上,下)半Fredholm算子的已有结果.     

The perturbation classes problem in Fredholm theory

We give a negative answer to the perturbation classes problem: the perturbation class of the upper semi-Fredholm operators contains properly the strictly singular operators, and the perturbation class of the lower semi-Fredholm operators contains properly the strictly cosingular operators.     

Stability of $$\mathcal{P}(S)$$ under Finite Perturbation

Abstract. T. Kato [9] found an important property of semi-Fredholm pencils, now called the Kato decomposition. Few years later, M.A. Kaashoek [7] introduced operators having the property as a generalization of semi-Fredholm operators. In [4], it is proved that these two notions are linked. The aim of this work is to study the stability of the property under finite perturbation.     

Preservation of closed sets under non-archimedean operators

In recent years some characterizations of the operators which preserve closed sets, closed bounded sets and closed bounded convex sets had been given in the literature. The kinds of operators involved are semi-Fredholm and tauberian operators. In this paper we introduce non-archimedean tauberian operators and prove that they coincide with semi-Fredholm operators when the ground field is spherically complete. Also, we give some theorems of preservation of closed sets under these classes of operators.     

S.G理论在研究Semi-Fredholm算子时的应用

本文研究了具有有限升标（降标）的半Fredholm算子，证明了具有有限 升标（降标）的上半Fredholm算子在其摄动类中交换元的摄动下仍具有同样性质，对于下半Fredholm算子有同样结论。从而改进了[1,2]中的主要结果。同时，我们证明了被摄动算子集合扩大（相对于[1]而言）而摄动仍为紧摄动时较[1]中更强的结果。     

The stability radius of a semi-Fredholm operator

We prove an asymptotic formula for the stability radius of a semi-Fredholm operator on a Banach space in terms of the reduced minimum modulus. In particular, this gives a new proof of the Fredholm case studied by Förster and Kaashoek [4].     

On the filling in holes problem for operator matrices

We consider upper-triangular 2-by-2 operator matrices and are interested in the set that has to be added to certain spectra of the matrix in order to get the union of the corresponding spectra of the two diagonal operators. We show that in the cases of the Browder essential approximate point spectrum, the upper semi-Fredholm spectrum, or the lower semi-Fredholm spectrum the set in question need not to be an open set but may be just a singleton. In addition, we modify and extend known results on Hilbert space operators to operators on Banach spaces.     

Some norm inequalities involving convex functions of operators

Positivity - We describe lower semi-Fredholm and Fredholm spectra of weighted composition operators on C(K) in the case when the corresponding map is an open map of the compact space K onto itself....     

ESSENTIAL APPROXIMATE POINT SPECTRA FOR UPPER TRIANGULAR MATRIX OF LINEAR RELATIONS

When A E ∈LR（H） and B E ∈LR（K） are given, for C E∈LR（K, H） we denoteby Mc the linear relation acting on the infinite dimensional separable Hilbert space H Kof the formIn this paper, we give the necessary and sufficient conditionson A and B for wh{ch Mc is upper semi-Fredholm with negative index or Weyl for some C C ∈LR（K, H）.     

On the bounded compact approximation property and measures of noncompactness

The Banach spaces with the bounded compact approximation property are characterized by the completeness of measures of noncompactness. Applications to the theory of semi-Fredholm operators and to the algebraic properties of their canonical images in the Calkin algebra are discussed.     

BOUNDED COMPOSITION OPERATORS ON HARDY SPACES (II)

In this paper, the characterization of semi-Fredholm composition operators from H2(Bm) to H2(Bn) is given. As an application of the composition operators, the extension properties of analytic maps are discussed.     

On additive maps preserving certain semi-Fredholm subsets

Let X and Y be two infinite dimensional real or complex Banach spaces, and let φ: ?(X)?→??(Y) be an additive surjective mapping that preserves semi-Fredholm operators in both directions. In the complex Hilbert space context, Mbekhta and ?emrl [M. Mbekhta and P. ?emrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), pp. 55–64] determined the structure of the induced map on the Calkin algebra. In this article, we show the following: given an integer n?≥?1, if φ preserves in both directions ? _n (X) (resp., 𝒬 _n (X)), the set of semi-Fredholm operators on X of non-positive (resp., non-negative) index, having dimension of the kernel (resp., codimension of the range) less than n, then φ(T)?=?UTV for all T or φ(T)?=?UT*V for all T, where U and V are two bijective bounded linear, or conjugate linear, mappings between suitable spaces.     

（1.2） INVERSES OF OPERATORS BETWEEN BANACH SPACES AND LOCAL CONJUGACY THEOREM

1.(1.2)InverseandLocalFinePropertyofaFamilyofOperatorsTxLetEandFbebothBanachspaces,andB(E,F)thesetofallboundedlinearoperatorsfromEintoFAnoperatorT B(F,E)issaidtobea(1.2)inverseofTifTT T=TandT TT =T .IfT satisfiesonlythefirstcondition,thenT issaidtobe...     

A RANK THEOREM FOR NONLINEAR SEMI-FREDHOLM OPERATORS BETWEEN TWO BANACH MANIFOLDS

In this article the concept of local conjugation of a C1 mapping between two Banach manifolds is introduced. Then a rank theorem for nonlinear semi-Fredholm operators between two Banach manifolds and a finite rank theorem are established in global analysis.