semi-Fredholm Operators Research Articles

Dimensions of subspaces of a Hilbert space and index of the semi-Fredholm operator

Let 2" denote the set of all closed subspaces of the Hilbert space H. The generalized dimension, dim gH0 for any , is introduced. Then an order is defined in [2H], the set of generalized dimensions of 2H. It makes [2H] totally ordered such that 0dim,H0dimg H for any Especially, a set of infinite dimensions are found out such that where m, n are integers with nm. Based on these facts, the generalized index, indg is defined for any A SF(H) (the set of all semi-Fredholm operators) and Tnd gA = is proved for any pure semi-Fredholm operator A SF+(H)(SF.(H)). The generalized index and dimension defined here are topologjcal and geometric, similar to the index of a Fredholm operator and the finite dimension. Some calculus of analysis can be performed on them (usually, and 1, 2,..., are identified with A known result deduced from this fact is not very proper, as will be shown later). For example, considering isometric operators in I (H) it is proved that V1,V2 are arcwise connected in B1x (H) (the set of all operators with left inverses) if and only if Ind, V1 = IndgV2. It follows that A, BeSF+(H)(SF_(H)) are arcwise connected in SF+(H)(SF_(H)) if and only if IndgAl=IndgB. The stability of Ind9 under compact or small perturbations and the continuity of the mapping Indg:SF(H) also hold. Thus the study of SF(H) is strictly based on geometric and analytic sense.

Semi-Fredholm operators with finite ascent or descent and perturbations

In this note we prove that the collection of upper (lower) semi-Fredholm operators with finite ascent (descent) is closed under commuting operator perturbations that belong to the perturbation class associated with the set of upper (lower) semi-Fredholm operators. Then, as a corollary we get the main result of S. Grabiner (Proc. Amer. Math. Soc.

Generalizations of semi-Fredholm operators

On nonseparable Hilbert spaces there are multiple sets of operators that are analogous to the semi-Fredholm operators on a separable space. We develop the properties of these sets and relate those properties to some recent research. We conclude with a theorem that indicates precisely how far one can go from a given generalized semi-Fredholm operator (or generalized Fredholm operator) and retain the property of generalized semi-Fredholmness (or generalized Fredholmness).

Generalizations of Semi-Fredholm Operators

On nonseparable Hilbert spaces there are multiple sets of operators that are analogous to the semi-Fredholm operators on a separable space. We develop the properties of these sets and relate those properties to some recent research. We conclude with a theorem that indicates precisely how far one can go from a given generalized semi-Fredholm operator (or generalized Fredholm operator) and retain the property of generalized semi-Fredholmness (or generalized Fredholmness).

Semi-Fredholm Operators with Finite Ascent or Descent and Perturbations

Semi-Fredholm Operators with Finite Ascent or Descent and Perturbations

Supertauberian operators and perturbations

Upper semi-Fredholm operators and tauberian operators in Banach spaces admit the following perturbative characterizations [6], [2]: An operator T: X --> Y is upper semi-Fredholm (tauberian) if and only if for every compact operator K: X --> Y the kernel N(T+K) is finite dimensional (reflexive). In [7] Tacon introduces an intermediate class between upper semi-Fredholm operators and tauberian operators, the supertauberian operators, and he studies this class using non-standard analysis. Here we study supertauberian operators using ultrapower of Banach spaces and, among other results, we obtain a perturbative characterization. As a consequence we characterize Banach spaces in which all superreflexive subspaces are finite dimensional, and Banach spaces in which all reflexive subspaces are superreflexive. Similar results are obtained for the dual class of cosupertauberian operators, including a perturbative characterization of this class, and characterizations of Banach spaces in which all quotients are finite dimensional, and Banach spaces in which all reflexive quotients are superreflexive.

On a Generalized Punctured Neighborhood Theorem in L (X)

Suppose that T is a bounded linear operator on a complex Banach space X. If T2(X) is closed, T(X) n N(T) is finite dimensional, and S is a bounded linear operator on X such that S is invertible, commutes with T, and has sufficiently small norm, then T S is upper semi-Fredholm. Throughout this paper X will denote a complex Banach space. We write Y(X) for the set of all bounded linear operators on X. For T E Y(X), we denote by N(T) the kernel and by T(X) the range of T. The operator T is called upper semi-Fredholm if T(X) is closed and dim N(T) < 00. We write a(T) for the spectrum of T. It is well known that the resolvent RA(T) = (AIT) -1 is a holomorphic function of A for points A in the resolvent set C a(T). The aim of this paper is the following generalization of the punctured neighborhood theorem for upper semi-Fredholm operators: Theorem 1. Suppose that T E A(X), T2 has closed range, and T(X) n N(T) is finite dimensional. Then: (a) T S is upper semi-Fredholm whenever S E S(X) is invertible, TS = ST, and 11S11 is sufficiently small. Furthermore, we have dim N(T S) = dim (N(T) nn Tn (X))a (b) If 0 is a boundary point of a(T), then 0 is a pole of the resolvent of T. For the proof of Theorem 1 we need some additional notation and a preliminary lemma. Let T E Y(X). We write a(T) and fl(T) for dim N(T) and codim T(X), respectively. The operator T is called lower semi-Fredholm if fl(T) is finite (in this case T has closed range, by [4, Satz 55.4]). T is called semi-Fredholm if T is upper or lower semi-Fredholm. T is Fredholm if both a(T) and ,@(T) are finite. The index of a semi-Fredholm operator T is defined by ind(T) = a(T) fl(T). Received by the editors November 9, 1992 and, in revised form, August 2, 1993. 1991 Mathematics Subject Classification. Primary 47A 10, 47A53, 47A55.

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].

Operational quantities characterizing semi-Fredholm operators

Several operational quantities have appeared in the literature characterizing upper semi-Fredholm operators. Here we show that these quantities can be divided into three classes, in such a way that two of them are equivalent if they belong to the same cla

The measure of noncompactness of Sobolev embeddings

New formulae for the upperq-norm of the embedding mapI from Sobolev spaces into Lebesgue spaces and, in particular, compactness criteria forI are given. Necessary and sufficient conditions for an operator to be a Φ+ (i.e., semi-Fredholm) operator are proved as well.

Approximation by Semi-Fredholm Operators

We present a simple approach for calculating some known distances from a bounded operator defined on a separable Hilbert space to certain sets related to semi-Fredholm operators.

Approximation by semi-Fredholm operators

We present a simple approach for calculating some known distances from a bounded operator defined on a separable Hilbert space to certain sets related to semi-Fredholm operators.

Robust Stability of Feedback Systems: A Geometric Approach Using the Gap Metric

A geometric framework for robust stabilization of infinite-dimensional time-varying linear systems is presented. The uncertainty of a system is described by perturbations of its graph and is measured in the gap metric. Necessary and sufficient conditions for robust stability are generalized from the time-invariant case. An example is given to highlight an important difference between the obstructions, which limit the size of a stabilizable gap ball, in the time-varying and time-invariant cases. Several results on the gap metric and the gap topology are established that are central in a geometric treatment of the robust stabilizability problem in the gap. In particular, the concept of a “graphable” subspace is introduced in the paper. Subspaces that fail to be graphable are characterized by an index condition on a certain semi-Fredholm operator.

Measures of noncompactness and upper semi-Fredholm perturbation theorems

We introduce the concept of a perturbation function, which allows us to give a general approach to the question of obtaining perturbation theorems for upper semi-Fredholm operators. Also, we show that the usual measures of noncompactness of continuous linear operators, as well as other related quantities, are perturbation functions.

A dense set of operators with tiny commutants

For a (bounded linear) operator T T on a complex, separable, infinite-dimensional Hilbert space H \mathcal {H} , let A ( T ) \mathcal {A}\,(T) and A a ( T ) {\mathcal {A}^a}(T) denote the weak closure of the polynomials in T T and, respectively, the weak closure of the rational functions with poles outside the spectrum of T T . Let A ′ ( T ) \mathcal {A}’(T) and A ( T ) \mathcal {A}(T) denote the commutant and, respectively, the double commutant of T T . We say that T T has a tiny commutant if A ′ ( T ) = A a ( T ) \mathcal {A}’(T)= {\mathcal {A}^a}(T) . By constructing a large family of "models" and by using standard techniques of approximation, it is shown that T ∈ L ( H ) : T T \in \mathcal {L}\,(\mathcal {H}):T has a tiny commutant is norm-dense in the algebra L ( H ) \mathcal {L}\,(\mathcal {H}) of all operators acting on H \mathcal {H} . Other related results: Let Lat B \operatorname {Lat}\;\mathcal {B} denote the invariant subspace lattice of a subalgebra B \mathcal {B} of L ( H ) \mathcal {L}(\mathcal {H}) . For a Jordan curve γ ⊂ C \gamma \subset {\mathbf {C}} , let γ ^ \hat \gamma denote the union of γ \gamma and its interior; for T ∈ L ( H ) T \in \mathcal {L}\;(\mathcal {H}) , let ρ s − F ( T ) = { λ ∈ C : λ − T {\rho _{s - F}}\,(T)= \{ \lambda \in {\mathbf {C}}:\lambda - T is a semi-Fredholm operator, and let ρ s − F + ( T ) ( ρ s − F − ( T ) ) = { λ ∈ ρ s − F ( T ) : ind ( λ − T ) &gt; 0 ( &gt; 0 , resp.)} \rho _{s - F}^ + (T)(\rho _{s - F}^ - (T))= \{ \lambda \in {\rho _{s - F}}(T):{\text {ind}}(\lambda - T) &gt; 0\;(&gt; 0,{\text {resp.)\} }} . With this notation in mind, it is shown that { T ∈ L ( H ) : A ( T ) = A a ( T ) } − = { T ∈ L ( H ) : Lat A ( T ) = Lat A a ( T ) } − = { A ∈ L ( H ) {\{ T \in \mathcal {L}(\mathcal {H}):\mathcal {A}(T)= {\mathcal {A}^a}(T)\} ^ - } = {\{ T \in \mathcal {L}(\mathcal {H}):\operatorname {Lat}\;\mathcal {A}(T)= \operatorname {Lat}\;{\mathcal {A}^a}(T)\} ^ - }= \{ A \in \mathcal {L}(\mathcal {H}) if γ \gamma (Jordan curve) ⊂ ρ s − F ± ( A ) \subset \rho _{s - F}^ \pm (A) , then γ ^ ⊂ σ ( A ) } \hat \gamma \subset \sigma (A)\} ; moreover, { A ∈ L ( H ) \{ A \in \mathcal {L}(\mathcal {H}) : if γ \gamma (Jordan curve) ⊂ ρ s − F ± ( A ) \subset \rho _{s - F}^ \pm (A) , then ind ( λ − A ) {\text {ind}}(\lambda - A) is constant on γ ^ ∩ ρ s − F ( A ) } ⊂ { T ∈ L ( H ) : A ( T ) = A ′ ( T ) } − ⊂ { T ∈ L ( H ) : Lat A ( T ) = Lat A ′ ( T ) } ⊂ { A ∈ L ( H ) \hat \gamma \cap {\rho _{s - F}}(A)\} \subset {\{ T \in \mathcal {L}(\mathcal {H}):\mathcal {A}(T)= \mathcal {A}’(T)\} ^ - } \subset \{ T \in \mathcal {L}(\mathcal {H}):\operatorname {Lat}\;\mathcal {A}(T)= \operatorname {Lat}\;\mathcal {A}’(T)\} \subset \{ A \in \mathcal {L}(\mathcal {H}) : if γ \gamma (Jordan curve) ⊂ ρ s − F ± ( A ) \subset \rho _{s - F}^ \pm (A) , then γ ^ ∩ ρ s − F ( A ) ⊂ ρ s − F ± ( A ) } ⊂ { T ∈ L ( H ) : A ( T ) = A a ( T ) } \hat \gamma \cap {\rho _{s - F}}(A) \subset \rho _{s - F}^ \pm (A)\} \subset \{ T \in \mathcal {L}(\mathcal {H}):\mathcal {A}(T)= {\mathcal {A}^a}(T)\} . (The first and the last inclusions are proper.) The results also include a partial analysis of Lat A ( T ) \operatorname {Lat}\;\mathcal {A}(T) .

Some characterizations ofp-adic semi-Fredholm operators

In this paper we give several descriptions of padic semi-Fredholm operators in terms of the preservation of orthogonal sequences and compactoid sets.

Approximation by semi-Fredholm operators with fixed nullity

Approximation by semi-Fredholm operators with fixed nullity

Seminorms related to weak compactness and to Tauberian operators

Let E and F be Banach spaces. The semigroup Φ+(E,F) of semi-Fredholm operators consists of the bounded linear mappings E→F with closed image and finite-dimensional kernel. By a well known result of Yood we have that T∈Φ+(E,F) if and only if for any bounded set B⊂E the condition TB relatively compact implies that B is relatively compact. Lebow and Schechter[10] gave a quantitative version of the above qualitative characterization, namely the operator T belongs to Φ+(E,F) if and only if there is c ≥ 0 such thatfor all bounded B⊂E. Here γ is the well known Hausdorff measure of non-compactnesswith BE the closed unit ball of E.

Perturbations of semi-Fredholm operators with complemented range

The proof of the theorem is quite involved and takes more than two pages. The purpose of this note is to give brief and transparent proofs of more general results. Our proofs rely on well known perturbation theorems which appear in [2]. It is important to note that if R(A) is complemented in Y, then R(A) is closed. This was proved in [2], IV. 1.2, without any assumptions on the kernel of A. Recall that a bounded linear operator A is called semi-Fredholm if R(A) is closed and c~(A)=dim ker A, ~(A)=codim R(A) are not both infinite. The index (A) = c~(A)-/~ (A) can be ___ ~.

Unitary approximation

A unitary approximant for a bounded linear operator T on a separable Hilbert space H is a unitary operator on H of minimum distance from T. In this paper, unitary approximants for certain Fredholm and semi-Fredholm operators of nonzero index are explictly constructed. This completes the answer to the question: which Hilbert space operators have a unitary approximant? The answer is: any operator of index zero, and any operator of nonzero index whose distance from the unitary group is greater than one, and no others. This and related questions are also completely answered in an arbitrary von Neumann algebra. For each element of the algebra, the distance to the invertible group and to the unitary group of the algebra is computed: these distances are related. It is shown that an element has no unitary approximant in the algebra if and only if it has nonzero (relative) index and it is a limit of invertibles in the algebra.