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Let H and K be infinite dimensional separable complex Hilbert spaces and B(K, H) the algebra of all bounded linear operators from K into H. Let A ∈ B (H) and B ∈ B (K) . We denote by M C the operator acting on H ⊕ K of the form M C = A C 0 B . In this paper, we give necessary and sufficient conditions for M C to be an upper semi-Fredholm operator with n (M C) > 0 and ind (M C) < 0 for some left invertible operator C ∈ B (K , H) . Meanwhile, we discover the relationship between n (M C) and n(A) during the exploration. And we also describe all left invertible operators C ∈ B (K , H) such that M C is an upper semi-Fredholm operator with n (M C) > 0 and ind (M C) < 0 . [ABSTRACT FROM AUTHOR]

Axiomatic Fredholm theory in unital C*-algebras was established in [D. Kečkić, Z. Lazović, Acta Sci. Math., 83(3-4):629--655, 2017]. Following the pure algebraic approach by Kečkić and Lazović, in the author's paper [S. Ivković, Banach J. Math. Anal., 17:51, 2023] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*-algebras. However, recently, in [S. Ivković, arXiv:2306.01133] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and is equivalent to the algebraic approach by Kečkić and Lazović. In this paper, we extend further that new Hilbert-module approach from Fredholm theory to semi-Fredholm and semi-Weyl theory in unital C*-algebras. Hence, we provide new proofs to the results in [S. Ivković, Banach J. Math. Anal., 17:51, 2023].

AbstractFor a bounded linear operator, by local spectral theory methods, we study the property [Formula: see text] which means that the difference of the approximate point spectrum with the upper semi-Fredholm spectrum coincides with the set of all finite-range left poles. We will investigate this property under closed proper subspaces of [Formula: see text] also under the tensor product. In addition, the relationships of this property with other spectral properties are studied. Among others, we will obtain several characterizations for the operators that verify the property (bz) and show that the set of these operators is closed.

Axiomatic Fredholm theory in unital C*-algebras was established in [D. Kečkić, Z. Lazović, Acta Sci. Math., 83(3-4):629-655, 2017]. Following the pure algebraic approach by Keckic and Lazovic, in the author’s paper [S. Ivkovic, Banach J. Math. Anal., 17:51, 2023] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*- algebras. However, recently, in [S. Ivković, arXiv:2306.01133] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and is equivalent to the algebraic approach by Kečkić and Lazović. In this paper, we extend further that new Hilbert-module approach from Fredholm theory to semi-Fredholm and semiWeyl theory in unital C*-algebras. Hence, we provide new proofs to the results in [S. Ivkovic, Banach J. Math. Anal., 17:51, 2023]. [ABSTRACT FROM AUTHOR]

For a bounded linear operator, by local spectral theory methods, we study the property (b z) , which means that the difference of the approximate point spectrum with the upper semi-Fredholm spectrum coincides with the set of all finite-range left poles. We will investigate this property under closed proper subspaces of X , also under the tensor product. In addition, the relationships of this property with other spectral properties are studied. Among others, we will obtain several characterizations for the operators that verify the property (bz) and show that the set of these operators is closed. [ABSTRACT FROM AUTHOR]

Let be an infinite dimensional complex Banach space, and let be the algebra of all bounded linear operators on. In this paper, given any positive integer n, we provide a characterization of all bijective maps on preserving the difference of semi-Fredholm operators with nullity less than n, and describe completely the structure of these maps. [ABSTRACT FROM AUTHOR]

In this paper, we establish some spectral analysis in Fredholm theory involving the concept of a polynomially Riesz operators perturbation. Moreover, we apply the obtained results to analyze the incidence of some essential spectra of a perturbed operator via this kind of perturbation. Our approach allows us to investigate a new description in the theory of unbounded operator matrices defined with maximal domain via this kind of perturbation. Finally, we conclude by presenting some progress in the theory of polynomially operators and the theory of semi-Fredholm and semi-Browder operators. [ABSTRACT FROM AUTHOR]

We carry out characterizations with techniques provided by the local spectral theory of bounded linear operators T ∈ L(X), X infinite dimensional complex Banach space, which verify property (V...) introduced by Sanabria et al. [Open Math. 16(1) (2018), 289-297). We also carry out the study for polaroid operators and Drazin invertible operators that verify the property mentioned above. [ABSTRACT FROM AUTHOR]

An operator T acting on a Banach space X satisfies the property (UWΠ) if σa(T)∖ σSF+−$\begin{array}{} \sigma_{SF_{+}^{-}} \end{array} $(T) = Π(T), where σa(T) is the approximate point spectrum of T, σSF+−$\begin{array}{} \sigma_{SF_{+}^{-}} \end{array} $(T) is the upper semi-Weyl spectrum of T and Π(T) the set of all poles of T. In this paper we introduce and study two new spectral properties, namely (VΠ) and (VΠa), in connection with Browder type theorems introduced in [1], [2], [3] and [4]. Among other results, we have that T satisfies property (VΠ) if and only if T satisfies property (UWΠ) and σ(T) = σa(T).

Given a complex Banach space X, we investigate the stable character of the property (VE) for a bounded linear operator T:X→X, under commuting perturbations that are Riesz, compact, algebraic and hereditarily polaroid. We also analyze sufficient conditions that allow the transfer of property (VE) from the tensorial factors T and S to its tensor product.

In this paper, we study the stability under direct sums and restrictions of some strong variations of Weyl and Browder type theorems recently introduced in Rashid and Prasad (Asia-Eur J Math 8:14, 2015. 10.1142/S1793557115500126), Sanabria et al. (Rev Colomb Mat 51(2):153-171, 2017a, Acta Math Univ Comen (NS) 86(2):345-356, 2017b, Open Math 16(1):289-297, 2018). [ABSTRACT FROM AUTHOR]

Abstract It is well known that for every bounded operator A in L ( H ) $L(H)$ , there exists a compact operator K in K ( H ) $K(H)$ such that the Weyl spectrum σ W ( A ) $\sigma_{W}(A)$ of the operator A coincides with the spectrum σ ( A + K ) $\sigma(A+K)$ of the perturbed operator A. In this work, we show the extension of this relation by the use of Kato’s decomposition to the set of semi-Fredholm operators.

An operator T acting on a Banach space X satisfies the property (UWE) if σa(T)\σSF+- (T) = E(T), where σa(T) is the aproximate point spectrum of T, σSF+- (T) is the upper semi-Weyl spectrum of T and E(T) is the set of all eigenvalues of T that are isolated in the spectrum σ(T) of T. In this paper, we introduce and study two new spectral properties, namely (VE) and (VEa), in connection with Weyl type theorems. Among other results, we have that T satisfies property (VE) if and only if T satisfies property (UWE) and σ(T) = σa(T). [ABSTRACT FROM AUTHOR]

It is well known that for every bounded operator A in $L(H)$, there exists a compact operator K in $K(H)$ such that the Weyl spectrum $\sigma_{W}(A)$ of the operator A coincides with the spectrum $\sigma(A+K)$ of the perturbed operator A. In this work, we show the extension of this relation by the use of Kato's decomposition to the set of semi-Fredholm operators. [ABSTRACT FROM AUTHOR]

In this paper we study the semi-Fredholm property of band-dominated operators A and prove that it already implies the Fredholmness of A in all cases where this is not disqualified by obvious reasons. Moreover, this observation is applied to show that the Fredholmness of a band-dominated operator already follows from the surjectivity of all its limit operators. [ABSTRACT FROM AUTHOR]

X. Fang proved that the backward shift Samuel multiplicity of a semi-Fredholm operatorequals the dimension of the kernel of, when the absolute value ofis small enough whose proof depended heavily on algebraic techniques. In this note, we prove the corresponding conclusion quite directly by analysis method. [ABSTRACT FROM PUBLISHER]

We characterize the Fredholm composition operators on the small Bloch-type spaces $${\mathcal{B}^{\alpha}}$$, with 0 < α < 1, and give necessary and sufficient conditions for their semi-Fredholmness. For nontrivial composition operators the semi-Fredholmness is equivalent to the operator having a closed range, i.e. being bounded below. These properties have been characterized when the composition operator acts on the Bloch-type spaces $${\mathcal{B}^{\alpha}, \alpha \ge1}$$. We extend some of those results, and give a few characterizations, for the leftover case of small Bloch-type spaces. [ABSTRACT FROM AUTHOR]

The authors prove that an operator with the cellular indecomposable property has no singular points in the semi-Fredholm domain, by applying the 4 × 4 matrix model of semi-Fredholm operators due to Fang in 2004. This result fills a gap in the result of Olin and Thomson in 1984. [ABSTRACT FROM AUTHOR]

Abstract: We show that none of the operational quantities that have been introduced to characterize the upper or the lower semi-Fredholm operators is equivalent to the corresponding perturbation radius. [Copyright &y& Elsevier]

In this work, we present some results concerning the operators defined on various classes of exotic Banach spaces, containing in particular those studied respectively by V. Ferenczi [7, 8] and T. Gowers with B. Maurey [14, 15]. We show that, on hereditarily indecomposable or quotient hereditarily indecomposable Banach space X, the set of bounded Fredholm operators is dense in ℒ( X), this gives that the boundary of bounded Fredholm operators is nothing else but the ideal of strictly singular operators if X is hereditarily indecomposable Banach space (resp. the ideal of strictly cosingular operators if X is quotient hereditarily indecomposable Banach space). On the other hand, a comparison between sufficiently rich and exotic Banach spaces is given via some properties of the two maps spectra and Wolf essential spectra. [ABSTRACT FROM AUTHOR]

The perturbation classes problem for semi-Fredholm operators asks when the equalities $${\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}$$ and $${\mathcal{SC}(X,Y)=P\Phi_-(X,Y)}$$ are satisfied, where $${\mathcal{SS}}$$ and $${\mathcal{SC}}$$ denote the strictly singular and the strictly cosingular operators, and PΦ and PΦ denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, $${\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)}$$ if and only if $${\mathcal{SC}(X,Y)=P\Phi_-(X,Y),}$$ and $${\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)}$$ if and only if $${\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}$$. Moreover we give examples showing that both direct implications fail in general. [ABSTRACT FROM AUTHOR]

We prove that the component PΦ+(X, Y ) of the perturbation class for the upper semi-Fredholm operators between Banach spaces X and Y coincide with the strictly singular operators when every closed infinite dimensional subspace of X contains an infinite dimensional complemented subspace whose complement is isomorphic to X. Similarly, we prove that the component PΦ•(X, Y ) of the perturbation class for the lower semi-Fredholm operators coincide with the strictly cosingular operators when every infinite codimensional subspace of Y is contained in an infinite codimensional complemented subspace isomorphic to Y . We also give examples of Banach spaces satisfying the aforementioned conditions. [ABSTRACT FROM AUTHOR]

Let ℒ(B(H)) be the algebra of all linear operators on B(H) and P be a property on B(H). For Φ1, Φ2 2 L(B(H)), we say that Φ1ΦP Φ2, whenever Φ1(T) has property P, if and only if Φ2(T) has this property. In particular, if I is the identity map on B(H), then Φ~P I means that Φ preserves property P in both directions. Each property P produces an equivalence relation on L(B(H)). We study the relation between equivalence classes with respect to different properties such as being Fredholm, semi-Fredholm, compact, finite rank, generalized invertible, or having a specific semi-index. [ABSTRACT FROM AUTHOR]

Abstract: We study the perturbation class for the upper semi-Fredholm operators when is non-empty. We show that it coincides with the strictly singular operators when and Y satisfies certain condition, and when and Y is arbitrary. As a consequence, we derive similar results for when . [Copyright &y& Elsevier]

This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular, if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property ( δ) in a neighborhood of 0. In this case, ind( T) ≤ ind( S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators with Bishop’s property ( β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that, under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property plays an essential role in the preservation of the index in this context. [ABSTRACT FROM AUTHOR]

Let T,A be operators with domains Ɗ(T) ⊆ Ɗ(A) in a normed space X. The operator A is called T-bounded if ∥Ax∥ ≤ a∥x∥+b∥Tx∥ for some a, b ≥ 0 and all x ϵ Ɗ(T). If A has the Hyers-Ulam stability then under some suitable assumptions we show that both T and S := A+T have the Hyers-Ulam stability. We also discuss the best constant of Hyers-Ulam stability for the operator S . Thus we establish a link between T -bounded operators and Hyers-Ulam stability. [ABSTRACT FROM AUTHOR]

Let H be an infinite-dimensional separable complex Hilbert space and [image omitted] the algebra of all bounded linear operators on H. We characterize surjective linear maps [image omitted] preserving semi-Fredholm operators in both directions. As an application we substantially improve a recently obtained characterization of linear preservers of generalized invertibility (Mbekhta, M., Rodman, L. and Semrl, P., 2006, Linear maps preserving generalized invertibility, Integral Equations Operator Theory, 55, 93-109.). The new proof given in this note is not only more efficient, but also much shorter and simpler. [ABSTRACT FROM AUTHOR]

Abstract: When and are given, we denote by the operator acting on the infinite dimensional separable Hilbert space of the form . In this paper, we prove that where , , , and denote the Browder spectrum, Browder essential approximate point spectrum, nullity, deficiency and conjugate of T, respectively. Some related results are obtained. [Copyright &y& Elsevier]

Abstract: When and are given, we denote by the operator acting on the infinite dimensional separable Hilbert space of the form . In this paper, it is shown that if A is upper semi-Fredholm of finite ascent and infinite codimension, and if is closed of infinite kernel, then is upper semi-Fredholm of finite ascent for some . In addition, we explore the hypercyclicity and supercyclicity for 2×2 upper triangular operator matrices on the Hilbert space. [Copyright &y& Elsevier]

When A ∈ B(H) and B ∈ B(K) are given, we denote by M C an operator acting on the Hilbert space H ⊕ K of the form $$ M_{C} = {\left( {\begin{array}{*{20}c} {A} & {C} \\ {0} & {B} \\ \end{array} } \right)} .$$ In this paper, first we give the necessary and sufficient condition for M C to be an upper semi-Fredholm (lower semi–Fredholm, or Fredholm) operator for some C ∈ B(K,H). In addition, let $$ \sigma _{{SF_{ + } }} $$ ( A) ={λ ∈ ℂ : A − λI is not an upper semi-Fredholm operator} be the upper semi–Fredholm spectrum of A ∈ B(H) and let $$ \sigma _{{SF_{ - } }} $$ ( A) = {λ ∈ ℂ : A − λI is not a lower semi–Fredholm operator} be the lower semi–Fredholm spectrum of A. We show that the passage from $$ \sigma _{{SF_{ ±} }} {\left( A \right)} \cap \sigma _{{SF_{ ±} }} {\left( B \right)}\;{\text{to}}\;\sigma_{{SF_{±} }} {\left( {M_{C} } \right)} $$ is accomplished by removing certain open subsets of $$ \sigma _{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)} $$ from the former, that is, there is an equality where $${\fancyscript G}$$ is the union of certain of the holes in $$ \sigma_{{SF_{± } }} {\left( {M_{C} } \right)} $$ which happen to be subsets of $$ \sigma_{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)} .$$ Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl's theorem, Browder's theorem, a–Weyl's theorem and a–Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space. [ABSTRACT FROM AUTHOR]

Abstract: We study the Fredholm theory for pairs of closed subspaces of a Banach space developed by Kato. We define the strictly singular and the strictly cosingular pairs of subspaces, and we show that some of the results of operator theory can be extended to this context. However, there are some notable differences. On the one hand, the perturbation classes problem has a positive answer in this context: the upper and lower semi-Fredholm pairs are stable under strictly singular and strictly cosingular perturbations, respectively, and this stability characterizes the strictly singular and the strictly cosingular pairs. Note that it has been proved recently that the perturbation classes problem for continuous semi-Fredholm operators has a negative answer. On the other hand, unlike in the case of operators, the Fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs. We also show the stability under composition of the compact, the strictly singular and the strictly cosingular pairs of subspaces. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

Given a complex Banach space X , we investigate the stable character of the property (V E) for a bounded linear operator T : X → X , under commuting perturbations that are Riesz, compact, algebraic and hereditarily polaroid. We also analyze sufficient conditions that allow the transfer of property (V E) from the tensorial factors T and S to its tensor product. [ABSTRACT FROM AUTHOR]