Browder Spectrum Research Articles

Locating the spectrum of a relation matrix through those of its inputs

Let [Formula: see text] and [Formula: see text] be Banach spaces. When [Formula: see text] and [Formula: see text] are linear relations in [Formula: see text] and [Formula: see text], respectively, we denote by [Formula: see text] the linear relation matrix acting on [Formula: see text] of the form [Formula: see text], where [Formula: see text] is the zero operator and [Formula: see text] is a bounded operator from [Formula: see text] to [Formula: see text]. In this paper, we prove that if [Formula: see text] denotes the Weyl spectrum, the Browder spectrum or the Drazin spectrum of a linear relation, then for every [Formula: see text] we have the equality [Formula: see text] where [Formula: see text] a subset of [Formula: see text]. Moreover, we explore how Weyl’s theorem and Browder’s theorem hold for linear relation matrices.

Left and right Browder elements in Banach algebras

In this paper, we introduce the left (resp. right) Browder elements in a semiprime, complex and unital Banach algebra A. It is established that (i) a∈A is left Browder if and only if a is relatively regular and the left quasinilpotent part of a (or, the left hyperkernel of a) is of finite order; (ii) the spectral mapping theorem holds for the left Browder spectrum; (iii) the set of all left Browder elements is an open multiplicative semigroup; (iv) the left Browder spectrum is stable under commuting Riesz perturbations; (v) Riesz elements are characterized by means of the invariance of the left Browder spectrum under commuting perturbations; (vi) the Kato decomposition for left Browder elements in primitive C* algebras is provided. Similar results hold for the right Browder elements and the right Browder spectrum.

Browder spectra of closed upper triangular operator matrices

<abstract><p>Let $ T_{B} = \left[\begin{array}{ll} A & B \\ 0 & D \end{array}\right] $ be an unbounded upper operator matrix with diagonal domain, acting in $ \mathcal H \oplus\mathcal K $, where $ \mathcal H $ and $ \mathcal K $ are Hilbert spaces. In this paper, some sufficient and necessary conditions are characterized under which $ T_{B} $ is Browder (resp., invertible) for some closable operator $ B $ with $ \mathcal D(B)\supset \mathcal D(D) $. Further, a sufficient and necessary condition is given under which the Browder spectrum (resp., spectrum) of $ T_{B} $ coincides with the union of the Browder spectrum (resp., spectrum) of its diagonal entries.</p></abstract>

Property (Bv) and Tensor Product

In this article, a-Browder’s classical theorem is considered through the property (Bv), and we show that if two operators are norm equivalent, then property (Bv) holds for one if and only if it holds for the other. The necessary conditions for the difference of the spectrum and essential approximate point spectrum of the tensor product of two operators coincide with the product of differences between the spectrum and the essential approximate point spectrum of its two factors are investigated. We also discuss the necessary conditions for the tensor product of two operators to verify the property (Bv) and simultaneously give the equivalence between their various spectra with their Browder spectrum, likewise with the Drazin spectrum.

Continuity of spectra on class of p-wA(s,t) operators

In this paper, we show that spectrum, Weyl spectrum and Browder spectrum are continuous on the set of p-wA(s,t) operators with 0 < p ? 1 and 0 < s, t, s + t ? 1.

Continuity of the spectrum and Weyl type theorems for extension of 2-isometric operators

In this paper, we show that the spectrum, Weyl spectrum, and Browder spectrum are continuous on the set of all 2-quasi-2-isometric operators. Further, we show that k-quasi-2-isometric operator satisfies Bishop's property β. Moreover, we prove Weyl type theorems for f(dTS), where dTS denote the generalized derivation or the elementary operator with k-quasi-2-isometric operator entries T and S* and f ∈ H(σ(dTS)), the set of analytic functions which are defined on an open neighborhood of σ(dTS).

Spectral properties for some extensions of isometric operators

In this paper, we show that the spectrum, Weyl spectrum, and Browder spectrum are continuous on the set of all 2-isometric operators. We also prove that if T, \(S^{*}\) are 2-isometric and invertible isometric operators respectively and X is a Hilbert–Schmidt operator such that \(TX=XS\), then \(T^{*}X=XS^{*}\). Moreover, we show that every Riesz projection E with respect to a non-zero isolated spectral point \(\lambda \) of a 2-isometric operator T is self-adjoint and satisfies \({\mathcal {R}}(E) = {\mathcal {N}}(T-\lambda ) = {\mathcal {N}}(T- \lambda )^{*}\). Further, we show that quasi-2-isometric operator satisfies Bishop’s property (\(\beta \)). Finally, we prove Weyl type theorems for \(f(d_{TS})\), where \( d_{TS}\) denote the generalized derivation or the elementary operator with quasi-2-isometric operator entries T and \(S^{*}\) and \(f \in H(\sigma (d_{TS}))\), the set of analytic functions which are defined on an open neighborhood of \(\sigma (d_{TS})\).

R-Fredholm theory in ordered Banach algebras

The introduction of Fredholm theory relative to general unital homomorphisms $$T:A \rightarrow B$$ between Banach algebras A and B, which involves the study of Fredholm, Weyl and Browder elements, was due to Harte (Math Z 179:431–436, 1982), after which this investigation was continued by several authors. Motivated by results of Alekhno (Positivity 11:375–386, 2007, Positivity 13:3–20, 2009), the definitions of upper Weyl and upper Browder elements in an ordered Banach algebra (OBA) were given in Benjamin and Mouton (Quaest Math 39:643–664, 2016), thereby initiating the study of the interplay between Fredholm theory and ordering. Inspired by an indication that these elements could be useful in the context of OBAs, certain variants of the invertible, Fredholm, Weyl and Browder elements (namely the r-invertible, r-Fredholm, r-Weyl and r-Browder elements) were investigated in Benjamin et al. (Glasgow Math J, 2018. https://doi.org/10.1017/S0017089518000393) in the context of general Banach algebras. Continuing the development of Fredholm theory in ordered Banach algebras, we now introduce the notions of upper r-Weyl, contractive upper r-Weyl, upper r-Browder and contractive upper r-Browder elements in an OBA and, motivated by previous results, investigate conditions under which the positive almost r-invertible r-Fredholm elements will be contractive upper r-Browder. This also leads to the recovering (and strengthening) of certain results in Benjamin and Mouton (Positivity 21:575–592, 2017) regarding the upper Browder spectrum property.

Limit points for Browder spectrum of operator matrices

Let $$A\in \mathcal {B}(X)$$ and $$B\in \mathcal {B}(Y)$$, where X and Y are Banach spaces, and let $$M_{C}$$ be an operator acting on $$X\oplus Y$$ given by $$M_C=\begin{pmatrix} A &{} \quad C 0 &{} \quad B \end{pmatrix}$$. We investigate the limit point set of the Browder spectrum of $$M_{C}$$. It is shown that $$\begin{aligned} acc \sigma _{b}(M_C)\cup W_{acc \sigma _{b}}= acc \sigma _{b}(A)\cup acc \sigma _{b}(B) \end{aligned}$$where $$W_{acc \sigma _{b}}$$ is a subsets of $$ acc\sigma _{*}(B)\cap acc\sigma _{*}(A)$$ and a union of certain holes in $$ acc \sigma _{b}(M_C)$$. Furthermore, several sufficient conditions for $$acc\sigma _{b}(M_C)=acc\sigma _{b}(A)\cup acc\sigma _{b}(B)$$ holds for every $$C\in \mathcal {B}(Y,X)$$ are given.

On the essential spectrum of the diagonal of an operator

ABSTRACT Let T be a bounded linear operator on a Banach space X, and let M be a closed T-invariant subspace of X. In this paper, we investigate the relationships between the essential spectrum and the Browder spectrum of T, and those of the operators and , where is the restriction of T to M, and is the operator induced by T on the quotient space X/M.

Spectral mapping theorems for the upper Weyl and upper Browder spectra

We discuss the spectral mapping properties of the upper Weyl and upper Browder spectra of an arbitrary ordered Banach algebra element.

Fredholm theory in ordered Banach algebras

This paper illustrates some initial steps taken in the effort of unifying the theory of positivity in ordered Banach algebas (OBAs) with the general Fredholm theory in Banach algebras. We introduce here upper Weyl and upper Browder elements in an OBA relative to an arbitrary Banach algebra homomorphism and investigate the spectra corresponding to the sets of upper Weyl and upper Browder elements, which we shall refer to as the upper Weyl and upper Browder spectra, respectively.

The upper Browder spectrum property

In this note we continue our study on the upper Browder spectrum initiated in Benjamin and Mouton (Quaest. Math. 39(5), 2016). Recall that, for an element a of an ordered Banach algebra A and w.r.t. a Banach algebra homomorphism \(T: A \rightarrow B,\) we have inclusions $$\begin{aligned} \sigma (Ta) \subseteq \beta _T(a) \subseteq \beta _T^+(a) \subseteq \sigma (a), \end{aligned}$$ where \(\sigma (Ta),\) \(\beta _T(a),\) \( \beta _T^+(a)\) and \( \sigma (a)\) denote the Fredholm, Browder, upper Browder and (usual) spectra of a, respectively (Benjamin and Mouton in Quaest. Math. 39(5), 2016). This paper concerns the following natural question: given that the spectral radius of a positive element is not in the Fredholm spectrum of the element, when will it be outside the upper Browder spectrum of that element?

Essential, Weyl and Browder spectra of unbounded upper triangular operator matrices

This paper is concerned with the spectral properties of the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which the essential spectrum, the Weyl spectrum and the Browder spectrum of such operator matrix, respectively, coincide with the union of the essential spectrum, the Weyl spectrum and the Browder spectrum of its diagonal entries.

Weyl type theorems of [formula omitted] upper triangular operator matrices

Let MC=(AC0B) be an upper triangular operator matrix on the Hilbert space H⊕H, where H is a Hilbert space. In this paper, necessary and sufficient conditions for σ⁎(MC)=σ⁎(A)∪σ⁎(B) are obtained, where σ⁎ is the essential spectrum, the Weyl spectrum, the Browder spectrum, the essential approximate point spectrum and the Browder essential approximate point spectrum. For MC, we further investigate the relationship among Weyl type theorems such as Weyl's, a-Weyl's, Browder's and a-Browder's theorems.

Perturbation of Browder spectrum of upper triangular operator matrices

Let be a 2-by-2 upper triangular operator matrix acting on the Banach space or Hilbert space . For the most important spectra such as spectrum, essential spectrum and Weyl spectrum, the characterizations of the perturbations of on Banach space have been presented. However, the characterization for the Browder spectrum on the Banach space is still unknown. The goal of this paper is to present some necessary and sufficient conditions for to be Browder for some using an alternative approach based on matrix representation of operators and the ghost index theorem. Moreover, in the Hilbert space case, the characterizations of the Fredholm and invertible perturbations of Browder spectrum are also given.

THE SPECTRAL CONTINUITY OF ESSENTIALLY HYPONORMAL OPERATORS

If A is a unital Banach algebra, then the spectrum can be viewed as a function � : A ! S, mapping each T 2 A to its spectrum �(T), where S is the set, equipped with the Hausdorff metric, of all compact subsets of C. This paper is concerned with the continuity of the spectrumvia Browder's theorem. It is shown thatis continuous when � is restricted to the set of essentially hyponormal operators for which Browder's theorem holds, that is, the Weyl spectrum and the Browder spectrum coincide. Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of bounded linear operators acting on H. If T ∈ B(H) write σ(T) and σp(T) for the spectrum and the set of eigenvalues of T, respectively. Let S denote the set, equipped with the Hausdorff metric, of all compact subsets of C. If A is a unital Banach algebra, then the spectrum can be viewed as a function σ : A → S, mapping each T ∈ A to its spectrum σ(T). It is known that the function σ is upper semicontinuous and that in noncommutative algebras, σ does have points of discontinuity. J. Newburgh (17) gave the fundamental results on spectral continuity in general Banach algebras. J. Conway and B. Morrel (7) have undertaken a detailed study of spectral continuity in the case where the Banach algebra is B(H). It seems to be interesting and challenging to identify classes C of operators for which σ becomes continuous when restricted to C. The first result of this study is: σ is continuous on the set of normal operators. On the other hand, Newburgh's argument uses the fact that the inverses of normal resolvents are normaloid (cf. see Solution 105 of (10)) and this argument is extended to the set of hyponormal operators because the inverses of hyponormal resolvents are also hyponormal and hence normaloid.

Browder spectra of upper triangular matrix linear relations

Browder spectra of upper triangular matrix linear relations

A Note on Browder Spectrum

In this paper we supply a new proof that the Browder spectrum is the largest part of the spectrum that remains unchanged under compact perturbations from the commutant.

The Taylor–Browder Spectrum on Prime C*-Algebras

We provide a formula for the Taylor–Browder spectrum of a pair (L a , R b ) of left and right multiplication operators acting on a prime C*-algebra with non-zero socle. We also compute ascent and descent for multiplication operators on a prime ring, characterise Browder elements in a prime C*-algebra and discuss upper semicontinuity for the Browder spectrum.