On uniform continuity of the spectral radius in Banach algebras

We investigate relations between different forms of subadditivity and submultiplicativity of the spectral radius. In particular, we prove that if the spectral radius is uniformly continuous on a Banach algebra, then the algebra is commutative modulo the radical; this confirms a conjecture raised by the second author in [7].     

On Lipschitz continuity of the joint spectral radius

Exponential stability of a discrete linear inclusion is characterized by the value of the joint (or generalized) spectral radius. This quantity is locally Lipschitz continuous on the space of compact irreducible sets of matrices. We give a brief outline of the proof.     

On the continuity of the spectrum in certain Banach algebras

In this paper we describe some classes of linear operatorsTL(H) (mainly Toeplitz, Wiener-Hopf and singular integral) on a Hilbert spacesH such that the spectrum (T, L(H)) is continuous at the pointsT from these classes. We also describe some subalgebras of the algebras for which the spectrum (x,) becomes continuous at the pointsx when (x,) is restricted to the subalgebra . In particular, we show that the spectrum (x,) is continuous in Banach algebras with polynomial identities. Examples of such algebras are given.This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

On the continuity of the generalized spectral radius in max algebra

Given a bounded set Ψ of n×n non-negative matrices, let ρ(Ψ) and μ(Ψ) denote the generalized spectral radius of Ψ and its max version, respectively. We show that

     

On spectral and central states on Banach algebras

Acta Mathematica Hungarica -     

The continuity of derivations of Banach algebras

This paper investigates conditions on a semisimple Banach algebra $\text{U}$ and a Banach $\text{U}$-module $\text{M}$ which insure that every derivation from $\text{U}$ into $\text{M}$ is necessarily a bounded linear operator.     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

Weak spectral synthesis in commutative Banach algebras

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Generalizing the notion of spectral sets in Δ(A), the considerably larger class of weak spectral sets was introduced and studied in [C.R. Warner, Weak spectral synthesis, Proc. Amer. Math. Soc. 99 (1987) 244-248]. We prove injection theorems for weak spectral sets and weak Ditkin sets and a Ditkin-Shilov type theorem, which applies to projective tensor products. In addition, we show that weak spectral synthesis holds for the Fourier algebra A(G) of a locally compact group G if and only if G is discrete.     

Every Banach algebra has the spectral radius property

Nylen and Rodman [NR] introduced the notion of spectral radius property in Banach algebras in order to generalize a classical theorem of Yamamoto on the asymptotic behaviour of the singular values of ann xn matrix. In this paper we prove a conjecture of theirs in the affirmative, namely that any unital Banach algebra has the spectral radius property. In fact a slightly more general spectral property holds. We show that for every element which has spectral points which are not of finite multiplicity, the essential spectral radius is the supremum of the set of absolute values of the spectral points that are not of finite multiplicity.     

Local spectral properties of multipliers on Banach algebras

Supported in part by NSF Grant DMS 90-96108. This work was completed while the first author visited Mississippi State University. Financial support and generous hospitality are gratefully acknowledged.     

On amalgamated Banach algebras

If F(z) is a polynomial of degree n having all zeros in \(|z|\le k,~k>0\) and f(z) is a polynomial of degree \(m\le n\) such that \(|f(z)|\le |F(z)|\) for \(|z|=k\), then it was formulated by Rather and Gulzar (Adv Inequal Appl 2:16–30, 2013) that for every \(|\delta |\le 1, |\beta |\le 1,~R>r\ge k\) and \(|z|\ge 1,\)

$$\begin{aligned} |B[fo\sigma ](z)+\psi B[fo\rho ](z)|\le |B[Fo\sigma ](z)+\psi B[Fo\rho ](z)|, \end{aligned}$$

where B is a \(B_{n}\) operator, \(\sigma (z){=}Rz, \rho (z){=}rz\) and \(\psi {:=}\psi (R,r,\delta ,\beta ,k) {=}\beta \bigg \{\bigg (\frac{R+k}{r+k}\bigg )^{n}{-}|\delta |\bigg \}{-}\delta \). The authors have assumed that \(B\in B_{n}\) is a linear operator which is not true in general. In this paper, besides discussing assumption of authors and their followers (see e.g, Rather et al. in Int J Math Arch 3(4):1533–1544, 2012), we present the correct proof of the above inequality. Moreover our result improves many prior results involving \(B_{n}\) operators and a number of polynomial inequalities can also be deduced by a uniform procedure.

     

Weak spectral synthesis in commutative Banach algebras. II

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Continuing our investigation in [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008) 987-1002], we establish various results on intersections and unions of weak spectral sets and weak Ditkin sets in Δ(A). As an important example, the algebra of n-times continuously differentiable functions is studied in detail. In addition, we prove a theorem on spectral synthesis for projective tensor products of commutative Banach algebras which applies to Fourier algebras of locally compact groups.     

B-fredholm and spectral properties for multipliers in Banach algebras

The main purpose of this paper is to study spectral and B-Fredholm properties of a multiplierT acting on a semi-simple regular tauberian commutative Banach algebraA. We show thatT is a B-Fredholm operator if and only ifT is a semi B-Fredholm operator, and in this case we have the indexind(T)=0. Moreover we give some spectral properties for multipliers. Spectral mapping theorems for the Weyl’s and B-Weyl spectrum of a multiplier are also considered. Furthermore we show that Weyl’s theorem and generalized Weyl’s theorem hold for a multiplierT. Finally we give sufficient conditions for a multiplier to be a product of an invertible and an idempotent operators.