A note on generalized Weyl's theorem

We prove that if either T or T^∗ has the single-valued extension property, then the spectral mapping theorem holds for B-Weyl spectrum. If, moreover T is isoloid, and generalized Weyl's theorem holds for T, then generalized Weyl's theorem holds for f(T) for every f∈H(σ(T)). An application is given for algebraically paranormal operators.     

Generalized Browder's and Weyl's theorems for Banach space operators

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem. We also prove that the spectral mapping theorem holds for the Drazin 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∈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∈H(σ(T)).     

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).     

Polaroid operators satisfying Weyl’s theorem

A Banach space operator T is polaroid and satisfies Weyl’s theorem if and only if T is Kato type at points λ ∈ iso σ(T) and has SVEP at points λ not in the Weyl spectrum of T. For such operators T, f(T) satisfies Weyl’s theorem for every non-constant function f analytic on a neighborhood of σ(T) if and only if f(T^∗) satisfies Weyl’s theorem.     

Topological uniform descent and Weyl type theorem

The generalized Weyl’s theorem holds for a Banach space operator T if and only if T or T^∗ has the single valued extension property in the complement of the Weyl spectrum (or B-Weyl spectrum) and T has topological uniform descent at all λ which are isolated eigenvalues of T. Also, we show that the generalized Weyl’s theorem holds for analytically paranormal operators.     

Inverse closedness of approximation algebras

We prove the inverse closedness of certain approximation algebras based on a quasi-Banach algebra X using two general theorems on the inverse closedness of subspaces of quasi-Banach algebras. In the first theorem commutative algebras are considered while the second theorem can be applied to arbitrary X and to subspaces of X which can be obtained by a general K-method of interpolation between X and an inversely closed subspace Y of X having certain properties. As application we present some inversely closed subalgebras of C(T) and C[−1,1]. In particular, we generalize Wiener's theorem, i.e., we show that for many subalgebras S of l¹(Z), the property {ck(f)}∈S (ck(f) being the Fourier coefficients of f) implies the same property for 1/f if f∈C(T) vanishes nowhere on T.     

A constructive proof for a theorem on contractive mappings in power sets

Recently I proved the following theorem: To every positive integer m there exists a positive integer h such that the following holds: If S is a set of h elements and f a mapping of the power set B of S into B such that f(T)?T for all T?B, then there exists a strictly increasing sequence T₁?…?T_m of subsets of S such that one of the following three possibilities holds: (a) All sets f(T_i), i=1,…,m, are equal. (b) For all i=1,…,m we have f(T_i)=T_i (c) T_i=f(T_i+1) for all i=1,…,m?1.The proof given in [2] was non-constructive. In this paper now we give a constructive proof. By the way, this also yields a solution of a problem of Rado [3, p. 106].     

On the Weyl Spectrum: Spectral Mapping Theorem and Weyl's Theorem

It is shown that ifTis a dominant operator or an analytic quasi-hyponormal operator on a complex Hilbert space and iffis a function analytic on a neighborhood of σ(T), then σ_w(f(T)) = f(σ_w(T)), where σ(T) and σ_w(T) stand respectively for the spectrum and the Weyl spectrum ofT; moreover, Weyl's theorem holds forf(T) + Fif “dominant” is replaced by “M-hyponormal,” whereFis any finite rank operator commuting withT. These generalize earlier results for hyponormal operators. It is also shown that there exist an operatorTand a finite rank operatorFcommuting withTsuch that Weyl's theorem holds forTbut not forT + F. This answers negatively a problem raised by K. K. Oberai (Illinois J. Math.21, 1977, 84–90). However, ifTis required to be isoloid, then the statement that Weyl's theorem holds forTwill imply it holds forT + F.     

On the distribution of the singular values of Toeplitz matrices

In 1920, G. Szegö proved a basic result concerning the distribution of the eigenvalues {λ⁽ⁿ⁾_j} of the Toeplitz sections T_n [f], where f(Θ)∈L_∞( -π, π) is a real-valued function. Simple examples show that this result cannot hold in the case where f(Θ) is not real valued. In this note, we give an extension of this theorem for the singular values of T_n[f] when f(Θ)=f₀(Θ)R₀(Θ) with f₀(Θ) real-valued and R₀(Θ) continuous, periodic (with period 2π) and such that |R₀(Θ)|=1. In addition, we apply the basic theorem of Szegö to resolve a question of C. Moler.     

Weyl's theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T∈B(X) is said to be totally hereditarily normaloid, T∈THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q?1, T∈H(q), if the quasi-nilpotent part H₀(T−λ)=(T−λ)^−q(0) for every complex number λ. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f(T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of σ(T), and T^∗ satisfies a-Weyl's theorem. If also T^∗ has the single valued extension property, then f(T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of σ(T) on which it is defined.     

Property (aw) and Weyl’s Theorem

In this note we study the property (aw), a variant of Weyl’s theorem introduced by Berkani and Zariouh, by means of the localized single valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (aw) holds. We also relate this property with Weyl’s theorem, a-Weyl’s theorem and property (w). Finally, we show that if T is a-polaroid and either T or T* has SVEP then f(T) satisfies property (aw) for each ${f \in H_1(\sigma(T))}$ .     

WEYL＇S TYPE THEOREMS AND HYPERCYCLIC OPERATORS

For a bounded operator T acting on an infinite dimensional separable Hilbert space H,we prove the following assertions: (i) If T or T* ∈ SC,then generalized aBrowder's theorem holds for f(T) for every ...     

Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ(fg) of the product of any two elements f and g in A coincides with the spectrum σ(TfTg). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and non-multiplicative unital map from a commutative C*-algebra into itself such that σ(TfTg)=σ(fg) holds for every f,g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ(TfTg)⊂σ(fg) holds for every f,g.     

Weyl’s Theorem for Functions of Operators and Approximation

Let H{\mathcal{H}} be a complex separable infinite dimensional Hilbert space. In this paper, we characterize those operators T on H{\mathcal{H}} satisfying that Weyl’s theorem holds for f(T) for each function f analytic on some neighborhood of σ(T). Also, it is proved that, given an operator T on H{\mathcal{H}} and ε > 0, there exists a compact operator K with ||K|| < e{\|K\| < \varepsilon} such that Weyl’s theorem holds for T + K.     

Weyl’s Theorem for Algebraically Quasi-class A Operators

If T or T* is an algebraically quasi-class A operator acting on an infinite dimensional separable Hilbert space then we prove that Weyl’s theorem holds for f(T) for every f ∈ H(σ(T)), where H(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is algebraically quasi-class A then a-Weyl’s theorem holds for f(T). Also, if T or T* is an algebraically quasi-class A operator then we establish that the spectral mapping theorems for the Weyl spectrum and the essential approximate point spectrum of T for every f ∈ H(σ(T)), respectively. This research was supported by the Kyung Hee University Research Fund in 2007 (KHU- 20071605).     

Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions

In this paper we prove that for an arbitrary pair {T ₁, T ₀} of contractions on Hilbert space with trace class difference, there exists a function ξ in L ¹(T) (called a spectral shift function for the pair {T ₁, T ₀}) such that the trace formula trace(f(T ₁) ? f(T ₀)) = ∫_T f′(ζ)ξ(ζ)dζ holds for an arbitrary operator Lipschitz function f analytic in the unit disk.     

Carleson and Vanishing Carleson Measures on Radial Trees

We extend a discrete version of an extension of Carleson’s theorem proved in [5] to a large class of trees that have certain radial properties. We introduce the geometric notion of s-vanishing Carleson measure on such a tree T (with s ≥ 1) and give several characterizations of such measures. Given a measure σ on T and p ≥ 1, let L ^p (σ) denote the space of functions g defined on T such that |g| ^p is integrable with respect to σ and let L ^p (? T) be the space of functions f defined on the boundary of T such that |f| ^p is integrable with respect to the representing measure of the harmonic function 1.We prove the following extension of the discrete version of a classical theorem in the unit disk proved by Power. A finite measure σ on T is an s-vanishing Carleson measure if and only if for any real number p > 1, the Poisson operator P : L ^p (? T) → L ^sp (σ) is compact. Characterizations of weak type for the case p = 1 and in terms of the tree analogue of the extended Poisson kernel are also given. Finally, we show that our results hold for homogeneous trees whose forward probabilities are radial and whose backward probabilities are constant, as well as for semihomogeneous trees.     

Generalized exponential dichotomy and global linearization

Hartman's linearization theorem says that if all eigenvalues of matrix A have no zero real part and f(x) is small Lipschitzian, then nonlinear system x^′=Ax+f(x) and its linear system x^′=Ax are topologically equivalent. In 1970s Palmer extended the theorem to nonautonomous systems. In this paper we extend Hartman's theorem to the systems with generalized exponential dichotomy.     

Weyl’s theorem for algebraically wF(p, r, q) operators with p, r?>?0 and q?≥?1

If T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1 acting in an infinite-dimensional separable Hilbert space, then we prove that Weyl’s theorem holds for f(T) for any f ∈ Hol(σ(T)), where Hol(σ(T)) is the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is a wF(p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl’s theorem holds for f(T). In addition, if T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1, then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum of T for any f ∈ Hol(σ(T)), respectively. Finally, we examine the stability of Weyl’s theorem and the a-Weyl’s theorem under commutative perturbations by finite-rank operators.     

Sard's theorem for mappings in Hölder and Sobolev spaces

We prove various generalizations of classical Sard's theorem to mappings f:M ^m →N ⁿ between manifolds in Hölder and Sobolev classes. It turns out that if f ∈ C ^k,λ (M ^m ,N ⁿ ), then—for arbitrary k and λ—one can obtain estimates of the Hausdorff measure of the set of critical points in a typical level set f ^?1(y). The classical theorem of Sard holds true for f ∈ C ^k with sufficiently large k, i.e., k>max(m?n,0); our estimates contain Sard's theorem (and improvements due to Dubovitskii and Bates) as special cases. For Sobolev mappings between manifolds, we describe the structure of f ^?1(y).