We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T. We also prove that the spectral mapping theorem holds for the B-Browder spectrum and for analytic functions on an open neighborhood of σ(T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f \in H(T), the space of functions analytic on an open neighborhood of σ(T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f \in H(σ(T)).
During the 1970s Brézis and Browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces. In this paper, we extend and refine their result to a general Banach space.
The main purpose of this paper, is to introduce and study the classes $(ab_{e})$ and $(aw_{e})$ which are strongly related to what has been recently studied in \cite{aznay-zariouh}. Furthermore, we give the connection between these classes and those that have been studied in \cite{berkani-zariouh0}. We also give an affirmative answer to a question asked in \cite{aznay-zariouh}.
Several spectra of analytically Riesz operators will be characterized. These results will led to prove Weyl and Browder type theorems for the aforementioned class of operators.
The main objective of this work is to study generalized Browder's and Weyl's theorems for the multiplication operators $L_A$ and $R_B$ and for the elementary operator $τ_{A,B}=L_AR_B$.
Let $M_{C}=\left(\begin{array}{cc}A&C\\0&B\\\end{array} \right)$ is a 2-by-2 upper triangular operator matrix acting on the Banach space $X\oplus Y$ or Hilbert space $H\oplus K$. For the most import spectra such as spectrum, essential spectrum and Weyl spectrum, the characterizations of the perturbations of $M_{C}$ 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 $M_C$ to be Browder for some $C\in B(Y,X)$ by using an alternative appr…
Given Banach spaces $\X$ and $\Y$ and Banach space operators $A\in L(\X)$ and $B\in L(\Y)$, let $ρ\colon L(\Y,\X)\to L(\Y,\X)$ denote the generalized derivation defined by $A$ and $B$, i.e., $ρ(U)=AU-UB$ ($U\in L(\Y,\X)$). The main objective of this article is to study Weyl and Browder type theorems for $ρ\in L(L(\Y,\X))$. To this end, however, first the isolated points of the spectrum and the Drazin spectrum of $ρ\in L(L(\Y,\X))$ need to be characterized. In addition, it will be also proved that if $A$ and $B$ are polaroid (respectively isoloid), then $ρ$ is polaroid (respectively isoloid).
In this paper, we show how the Brezis-Browder theorem for maximally monotone multifunctions with a linear graph on a reflexive Banach space, and a consequence of it due to Yao, can be generalized to SSDB spaces.
Let $T$ be a bounded linear operator on a Banach space $X$. We give new necessary and sufficient conditions for $T$ to be Drazin or Koliha-Drazin invertible. All those conditions have the following form: $T$ possesses certain decomposition property and zero is not an interior point of some part of the spectrum of $T$. In addition, we study operators $T$ satisfying Browder\textquoteright s theorem, or a-Browder\textquoteright s theorem, by means of some relationships between diferent parts of the spectrum of $T$.
In this note, we study maximal monotonicity of linear relations (set-valued operators with linear graphs) on reflexive Banach spaces. We provide a new and simpler proof of a result due to Brezis-Browder which states that a monotone linear relation with closed graph is maximal monotone if and only if its adjoint is monotone. We also study Fitzpatrick functions and give an explicit formula for Fitzpatrick functions of order n for monotone symmetric linear relations.
The transfer property for the generalized Browder's theorem both of the tensor product and of the left-right multiplication operator will be characterized in terms of the $B$-Weyl spectrum inclusion. In addition, the isolated points of these two classes of operators will be fully characterized.
Given two complex Banach spaces $X_1$ and $X_2$, a tensor product of $X_1$ and $X_2$, $X_1\tilde{\otimes}X_2$, in the sense of J. Eschmeier ([5]), and two finite tuples of commuting operators, $S=(S_1,\ldots ,S_n)$ and $T=(T_1,\ldots ,T_m)$, defined on $X_1$ and $X_2$ respectively, we consider the $(n+m)$-tuple of operators defined on $X_1\tilde{\otimes}X_2$, $(S\otimes I,I\otimes T)= (S_1\otimes I,\ldots ,S_n\otimes I,I\otimes T_1,\ldots ,I \otimes T_m)$, and we give a description of the semi-Browder joint spectra introduced by V. Kordula, V. Müller and V. Rako$\check{c}$evi$\acute{ c}$ in [7…
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 $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}$. We investigate the limit point set of the Browder spectrum of $M_{C}$. It is shown that $$ acc σ_{b}(M_C)\cup W_{acc σ_{b}}= acc σ_{b}(A)\cup acc σ_{b}(B)$$ where $W_{acc σ_{b}}$ is a subsets of $ accσ_{*}(B)\cap accσ_{*}(A)$ and a union of certain holes in $ acc σ_{b}(M_C)$. Furthermore, several sufficient conditions for $accσ_{b}(M_C)=accσ_{b}(A)\cup accσ_{b}(B)$ holds for every $C\in \ma…
A Banach space operator $T\in B({\cal X})$ is polaroid if points $λ\in\isoσσ(T)$ are poles of the resolvent of $T$. Let $σ_a(T)$, $σ_w(T)$, $σ_{aw}(T)$, $σ_{SF_+}(T)$ and $σ_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $C\in B({\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $π_0(M_C)=\{λ\in\isoσ(M_C) 0<\dim(M_C-λ)^{-1}(0)<\infty\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$…
We characterize the properties $(z)$ and $(az)$ for an operator $T$ whose dual $T^*$ has the SVEP on the complementary of the upper semi-Weyl spectrum of $T.$ If $S$ and $T$ are Banach space operators satisfying property $(z)$ or $(az),$ we give conditions on $S$ and $T$ to ensure the preservation of these properties by the direct sum $S\oplus T.$ Some results are given for multipliers and in general for $(H)$-operators. Also we give a correct proof of \cite[Theorem 2.3]{SZ} which was proved by using the equality $σ_p^0(S\oplus T)= σ_p^0(S)\cup σ_p^0(T).$ However this equality is not true; we …
In this paper, we define a Banach SNL space to be a Banach space with a certain kind of linear map from it into its dual, and we develop the theory of linear $L$-positive subsets of Banach SNL spaces with Banach SNL dual spaces. We use this theory to give simplified proofs of some recent results of Bauschke, Borwein, Wang and Yao, and also of the classical Brezis-Browder theorem.
Let ${\bf R}$ denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator $T$ on a Banach space $X$ we prove that $T=T_M\oplus T_N$ with $T_M \in {\bf R}$ and $T_N$ quasinilpotent (nilpotent) if and only if $T$ admits a generalized Kato decomposition ($T$ is of Kato type) and $0$ is not an interior point of the corresponding spectrum $σ_{\bf R}(T)=\{λ\in \mathbb{C}: T-λ\notin {\bf R}\}$. In addition, we show that every non…
We show that a variant of a function defined by Cholamjiak and Suantai can be used to characterize 2-uniform smoothness. We then obtain greatly simplified proofs of a convergence theorem of Marino and Xu and of one of Zhou using a generalization of a lemma of Browder and Petryshyn and the aforementioned characterization. We also obtain more easily rates of asymptotic regularity corresponding to the studied iterations. Finally, we derive a way to relate two constants which are characteristic to 2-uniformly smooth spaces.
We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.
In the setting of hyperbolic spaces, we show that the convergence of Browder-type sequences and Halpern iterations respectively entail the convergence of their viscosity version with a Rakotch map. We also show that the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type sequence. Our results follow from proof-theoretic techniques (proof mining). From an analysis of theorems due to T. Suzuki, we extract a transformation of rates for the original Browder type and Halpern iterations into rates for the corresponding viscos…
The paper introduces the notion of properties $(Z_{Π_a})$ and $(Z_{E_a})$ as variants of Weyl's theorem and Browder's theorem for bounded linear operators acting on infinite dimensional Banach spaces. A characterization of these properties in terms of localized single valued extension property is given, and the perturbation by commuting Riesz operators is also studied. Classes of operators are considered as illustrating examples.
In this paper, we define an analytical index for continuous families of Fredholm operators parameterized by a topological space $\mathbb{X}$ into a Banach space $X.$ We also consider the Weyl spectrum for continuous families of bounded linear operators and we study its continuity.
We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concepts of "$r_L$-density" and its specialization "quasidensity": the closed quasidense monotone multifunctions from a Banach space into its dual form a (generally) strict subset of the maximally monotone ones, though all surjective maximally monotone and all maximally monotone multifunctions on a reflexive space are quasidense. W…
We give necessary and sufficient conditions for a Banach space operator with the single valued extension property (SVEP) to satisfy Weyl's theorem and $a$-Weyl's theorem. We show that if $T$ or $T^{\ast}$ has SVEP and $T$ is transaloid, then Weyl's theorem holds for $f(T)$ for every $f\in H(σ(T))$. When $T^{\ast}$ has SVEP, $T$ is transaloid and $T$ is $a$-isoloid, then $a$-Weyl's theorem holds for $f(T)$ for every $f\in H(σ(T))$. We also prove that if $T$ or $T^{\ast}$ has SVEP, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.
Muraleetharan and Thirulogasanthan in (J. Math. phys. 59, No. 10, 103506, 27p. (2018)) introduced the concept of Calkin Sspectrum of a bounded quaternionic linear operators. The study of this spectrum is establisched using the Fredholm operators theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two elements in quaternionic Banach algebra by means of the Fredholm S-spectrum of the two elements. Next, we prove a perturbation result on this spectr…
For $n$-normal operators $A$ [2, 4, 5], equivalently $n$-th roots $A$ of normal Hilbert space operators, both $A$ and $A^*$ satisfy the Bishop--Eschmeier--Putinar property $(β)_ε$, $A$ is decomposable and the quasi-nilpotent part $H_0(A-λ)$ of $A$ satisfies $H_0(A-λ)^{-1}(0)=(A-λ)^{-1}(0)$ for every non-zero complex $λ$. $A$ satisfies every Weyl and Browder type theorem, and a sufficient condition for $A$ to be normal is that either $A$ is dominant or $A$ is a class ${\mathcal A}(1,1)$ operator.
In this paper, we analyze multi-dimensional Weyl almost periodic type functions in Lebesgue spaces with variable exponents. The introduced classes seem to be new and not considered elsewhere even in the constant coefficient case. We provide certain applications to the abstract Volterra integro-differential equations in Banach spaces.
Let $A\in\mathcal{B}(X)$, $B\in\mathcal{B}(Y)$ and $C\in\mathcal{B}(Y,X)$ where $X$ and $Y$ are infinite Banach or Hilbert spaces. Let $M_{C}=\begin{pmatrix} A & C\cr 0 & B \end{pmatrix}$ be $2\times 2$ upper triangular operator matrix acting on $X\oplus Y$. In this paper, we consider some necessary and sufficient conditions for $M_{C}$ to be generalized Drazin-Riesz invertible. Furthermore, the set $\bigcap_{C\in \mathcal{B}(Y,X)}σ_{gDR}(M_{C})$ will be investigated and their relation between $\bigcap_{C\in \mathcal{B}(Y,X)}σ_{b}(M_{C})$ will be studied, where $σ_{gDR}(M_{C})$ and $σ_{b}(M_{C…
We describe a construction of wavelets (coherent states) in Banach spaces generated by ``admissible'' group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example we consider operator valued Segal-Bargmann type spaces and the Weyl functional calculus. Keywords: Wavelets, coherent states, Banach spaces, group representations, covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann spaces, Weyl functional calculus (quantization), second quantization, bosonic field.
Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the poles $Π(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the approximate point spectrum and the left ploles $Π^a(A)$) of the spectrum of $A$ satisfy $Π(A)\subseteq E(A)$ (resp., $Π^a(A)\subseteq E^a(A)$), and the reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$. Evidently $Π(A)\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and $Π^a(A)$. The study of identities $E(A)=Π^a(A)$ and $E^a(A)=Π(A)$, an…
In this paper, we introduce and analyze several different notions of Weyl almost periodic functions and Weyl ergodic components in Lebesgue spaces with variable exponent $L^{p(x)}.$ We investigate the invariance of (asymptotical) Weyl almost periodicity with variable exponent under the actions of convolution products, providing also some illustrative applications to abstract fractional differential inclusions in Banach spaces. The introduced classes of generalized (asymptotically) Weyl almost periodic functions are new even in the case that the function $p(x)$ has a constant value $p\geq 1,$ p…
In \cite{koliha}, Koliha proved that $T\in L(X)$ ($X$ is a complex Banach space) is generalized Drazin invertible operator equivalent to there exists an operator $S$ commuting with $T$ such that $STS = S$ and $σ(T^{2}S - T)\subset\{0\}$ which is equivalent to say that $0\not\in \mbox{acc}\,σ(T).$ Later, in \cite{rwassa,rwassa1} the authors extended the class of generalized Drazin invertible operators and they also extended the class of pseudo-Fredholm operators introduced by Mbekhta \cite{mbekhta} and other classes of semi-Fredholm operators. As a continuation of these works, we introduce and …
Let $X$ be a real locally uniformly convex Banach space and $X^*$ be the dual space of $X$. Let $φ:\mathbf R_+\to \mathbf R_+$ be a strictly increasing and continuous function such that $φ(0) = 0$, $φ(r) \to \infty$ as $r\to\infty$, and let $J_φ$ be the duality mapping corresponding to $φ$. We will prove that for every $R>0$ and every $x_0\in X$ there exists a nondecreasing function $ψ= ψ(R, x_0) :\mathbf R_+\to \mathbf R_+$ such that $ψ(0) = 0$, $ψ(r)>0$ for $r>0$, and $\langle x^*- x_0^*, x-x_0\rangle \ge ψ(\|x-x_0\|) \|x-x_0\|$ for all $x$ satisfying $\|x-x_0\|\le R$ and all $x^*\in J_φx$ a…
Let $X$ be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra $\ell^{1}(\mathbb{N}_0)$ and the algebraic structure of Cesàro sums of a linear operator $T\in \mathcal{B}(X)$ is established. In particular, we show that every $(C, α)$-bounded operator $T$ induces - and is in fact characterized - by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for A…
An operator $T \in \mathcal{B}(X)$ defined on a Banach space $X$ satisfies property $(gb)$ if the complement in the approximate point spectrum $σ_{a}(T)$ of the upper semi-B-Weyl spectrum $σ_{SBF_{+}^{-}}(T)$ coincides with the set $Π(T)$ of all poles of the resolvent of $T$. In this note we continue to study property $(gb)$ and the stability of it, for a bounded linear operator $T$ acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators commuting with $T$. Two counterexamples show that property $(gb)$ in general is not prese…
A bounded linear operator $T$ on a Banach space $X$ is said to be generalized Drazin-meromorphic invertible if there exists a bounded linear operator $S$ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is meromorphic. We shall say that $T$ admits a generalized Kato-meromorphic decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and the reduction $T_N$ is meromorphic. In this paper we shall investigate such kind of operators and corresponding spectra, the generalized Drazin-meromorphic spectrum and the generalize…
We systematize and generalize recent results of Gerlach and Glück on the strong convergence and spectral theory of bounded (positive) operator semigroups $(T_s)_{s\in S}$ on Banach spaces (lattices). (Here, $S$ can be an arbitrary commutative semigroup, and no topological assumptions neither on $S$ nor on its representation are required.) To this aim, we introduce the "semigroup at infinity" and give useful criteria ensuring that the well-known Jacobs--de Leeuw--Glicksberg splitting theory can be applied to it. Next, we confine these abstract results to positive semigroups on Banach lattices w…
Let $X$ a Banach space and $T$ a bounded linear operator on $X.$ We denote by $S(T)$ the set of all $λ\in \cit$ such that $T$ does not have the single-valued extension property at $λ$. In this note we prove equality up to $S(T)$ between the left Drazin spectrum and the left B-Fredholm spectrum and between the semi-essential approximate point spectrum and the left Drazin spectrum. As applications we investigate generalized Weyl's theorem for operator matrices and multipliers operators.
This paper is concerned with frame decompositions of $α$-modulation spaces. These spaces can be obtained as coorbit spaces for square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. The theory yields canonical constructions for Banach frames or atomic decompositions in $α$-modulation spaces. A necessary ingredient in this abstract machinery is the existence of generating functions that are admissible for the representation. For numerical purposes, admissible atoms with compact support are necessary. We show new admissibility conditions that considerabl…
We shall say that a bounded linear operator $T$ acting on a Banach space $X$ admits a generalized Kato-Riesz decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and $T_N$ is Riesz. In this paper we define and investigate the generalized Kato-Riesz spectrum of an operator. For $T$ is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator $S $ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is Riesz. We investigate generalized Drazin-Riesz invertible operators and also, characteri…
Classical pseudo-differential calculus on $\mathbb{R}^{d}$ can be viewed as a (non-commutative) functional calculus for the standard position and momentum operators $(Q_{1}, \dots , Q_{d})$ and $(P_{1}, \dots , P_{d})$. We generalise this calculus to the setting of two $d$-tuples of operators $A=(A_{1}, \dots , A_{d})$ and $B=(B_{1}, \dots , B_{d})$ acting on a Banach space $X$ such that $iA_{1}, \dots , iA_{d}$ and $iB_{1}, \dots , iB_{d}$ generate bounded $C_0$-groups satisfying the Weyl canonical commutation relations $e^{isA_j}e^{itA_k} = e^{itA_k}e^{isA_j}$, $e^{isB_j}e^{itB_k} = e^{itB_k…
By a theorem of Bayart, $φ$ generates a bounded composition operator on the Hardy space $\Hp$of Dirichlet series ($1\le p<\infty$) only if $φ(s)=c_0 s+ψ(s)$, where $c_0$ is a nonnegative integer and $ψ$ a Dirichlet series with the following mapping properties: $ψ$ maps the right half-plane into the half-plane $\Real s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\Hp$ are bounded below by a constant times $r^n$ for some $0<r<1$ when $c_0=0$ and …
