The norm spectrum in certain classes of commutative Banach algebras

H. S. Mustafayev

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Almost multiplicative functions on commutative Banach algebras

S. H. Kulkarni, D. Sukumar (2010)

Studia Mathematica

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Let A be a complex commutative Banach algebra with unit 1 and δ > 0. A linear map ϕ: A → ℂ is said to be δ-almost multiplicative if |ϕ(ab) - ϕ(a)ϕ(b)| ≤ δ||a|| ||b|| for all a,b ∈ A. Let 0 < ϵ < 1. The ϵ-condition spectrum of an element a in A is defined by $σ_{ϵ} (a) : = λ \in ℂ : | | λ - {a | | | | (λ - a)}^{- 1} | | \geq 1 / ϵ$ with the convention that $| | λ - {a | | | | (λ - a)}^{- 1} | | = \infty$ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If ϕ(1) = 1 and ϕ is δ-almost multiplicative, then $ϕ (a) \in σ_{δ} (a)$ for all a in A. (2) If ϕ is linear and $ϕ (a) \in σ_{ϵ} (a)$ for...

Isometries between groups of invertible elements in Banach algebras

Osamu Hatori (2009)

Studia Mathematica

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We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then $T {(1)}^{- 1} T$ is an isometrical group isomorphism. In particular, $T {(1)}^{- 1} T$ extends to an isometrical real algebra isomorphism from A onto B.

(Non-)amenability of ℬ(E)

Volker Runde (2010)

Banach Center Publications

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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ (E) = (E) + ℂ i d_{E}$ . Still, ℬ(ℓ²) is...

Algebra isomorphisms between standard operator algebras

Thomas Tonev, Aaron Luttman (2009)

Studia Mathematica

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If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ B(Y), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ is the set $σ_{π} (A) = λ \in σ (A) : | λ | = m a x_{z \in σ (A)} | z |$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation $σ_{π} (φ (A) \circ φ (B)) = σ_{π} (A B)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be...

The Banach algebra of continuous bounded functions with separable support

M. R. Koushesh (2012)

Studia Mathematica

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We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_{s} (X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if...

Fermi Golden Rule, Feshbach Method and embedded point spectrum

Jan Dereziński (1998-1999)

Séminaire Équations aux dérivées partielles

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A method to study the embedded point spectrum of self-adjoint operators is described. The method combines the Mourre theory and the Limiting Absorption Principle with the Feshbach Projection Method. A more complete description of this method is contained in a joint paper with V. Jak $\overset{ˇ}{s}$ ić, where it is applied to a study of embedded point spectrum of Pauli-Fierz Hamiltonians.

On the functional equation defined by Lie's product formula

Gerd Herzog, Christoph Schmoeger (2006)

Studia Mathematica

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Let E be a real normed space and a complex Banach algebra with unit. We characterize the continuous solutions f: E → of the functional equation $f (x + y) = l i m_{n \to \infty} (f (x / n) f (y / n)) ⁿ$ .

The third order spectrum of the p-biharmonic operator with weight

Khalil Ben Haddouch, Najib Tsouli, Zakaria El Allali (2014)

Applicationes Mathematicae

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We show that the spectrum of $Δ ²_{p} u + {2 β \cdot \nabla (| Δ u |}^{p - 2} {Δ u) + | β | ² | Δ u |}^{p - 2} Δ u = α m {| u |}^{p - 2} u$ , where $β \in ℝ^{N}$ , under Navier boundary conditions, contains at least one sequence of eigensurfaces.

Resonant delocalization for random Schrödinger operators on tree graphs

Michael Aizenman, Simone Warzel (2013)

Journal of the European Mathematical Society

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We analyse the spectral phase diagram of Schrödinger operators $T + λ V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by $i i d$ random variables. The main result is a criterion for the emergence of absolutely continuous $(a c)$ spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials $a c$ spectrum appears at arbitrarily weak disorder $(λ ≪ 1)$ in an energy regime which extends beyond the spectrum of $\sim T$ ....

Approximate amenability for Banach sequence algebras

H. G. Dales, R. J. Loy, Y. Zhang (2006)

Studia Mathematica

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We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where $A = ℓ^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras $ℓ^{p} (ω)$ .

Some examples of cocycles with simple continuous singular spectrum

K. Frączek (2001)

Studia Mathematica

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We study spectral properties of Anzai skew products $T_{φ} : ² \to ²$ defined by $T_{φ} (z, ω) = (e^{2 π i α} z, φ (z) ω)$ , where α is irrational and φ: → is a measurable cocycle. Precisely, we deal with the case where φ is piecewise absolutely continuous such that the sum of all jumps of φ equals zero. It is shown that the simple continuous singular spectrum of $T_{φ}$ on the orthocomplement of the space of functions depending only on the first variable is a “typical” property in the above-mentioned class of cocycles, if α admits a sufficiently...

On generalized property (v) for bounded linear operators

J. Sanabria, C. Carpintero, E. Rosas, O. García (2012)

Studia Mathematica

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An operator T acting on a Banach space X has property (gw) if $σ_{a} (T) ∖ σ_{S B F ₊ ¯} (T) = E (T)$ , where $σ_{a} (T)$ is the approximate point spectrum of T, $σ_{S B F ₊ ¯} (T)$ is the upper semi-B-Weyl spectrum of T and E(T) is the set of all isolated eigenvalues of T. We introduce and study two new spectral properties (v) and (gv) in connection with Weyl type theorems. Among other results, we show that T satisfies (gv) if and only if T satisfies (gw) and $σ (T) = σ_{a} (T)$ .

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F (x) : = F ̃ x^{(2 n + 1)}$ where $F ̃ : X^{2 n + 1} \to Y$ is a continuous (2n+1)-linear operator.

Fredholm spectrum and growth of cohomology groups

Jörg Eschmeier (2008)

Studia Mathematica

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Let T ∈ L(E)ⁿ be a commuting tuple of bounded linear operators on a complex Banach space E and let $σ_{F} (T) = σ (T) ∖ σ_{e} (T)$ be the non-essential spectrum of T. We show that, for each connected component M of the manifold $R e g (σ_{F} (T))$ of all smooth points of $σ_{F} (T)$ , there is a number p ∈ 0, ..., n such that, for each point z ∈ M, the dimensions of the cohomology groups $H^{p} ({(z - T)}^{k}, E)$ grow at least like the sequence ${(k^{d})}_{k \geq 1}$ with d = dim M.

On the convergence and character spectra of compact spaces

István Juhász, William A. R. Weiss (2010)

Fundamenta Mathematicae

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An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

On the existence of non-linear frames

Shah Jahan, Varinder Kumar, S.K. Kaushik (2017)

Archivum Mathematicum

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A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that $𝒳^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then $𝒳^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

A spectral mapping theorem for Banach modules

H. Seferoğlu (2003)

Studia Mathematica

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Let G be a locally compact abelian group, M(G) the convolution measure algebra, and X a Banach M(G)-module under the module multiplication μ ∘ x, μ ∈ M(G), x ∈ X. We show that if X is an essential L¹(G)-module, then $σ (T_{μ}) = \bar{μ ̂ (s p (X))}$ for each measure μ in reg(M(G)), where $T_{μ}$ denotes the operator in B(X) defined by $T_{μ} x = μ \circ x$ , σ(·) the usual spectrum in B(X), sp(X) the hull in L¹(G) of the ideal $I_{X} = f \in L ¹ (G) | T_{f} = 0$ , μ̂ the Fourier-Stieltjes transform of μ, and reg(M(G)) the largest closed regular subalgebra of M(G); reg(M(G)) contains...

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f : {[0, 1]}^{m} \to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f : {[0, 1]}^{m} \to X$ with respect to any norming subset there exists a separately increasing function $g : {[0, 1]}^{m} \to ℝ$ such that the sets of...

Conditions equivalent to C* independence

Shuilin Jin, Li Xu, Qinghua Jiang, Li Li (2012)

Studia Mathematica

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Let and be mutually commuting unital C* subalgebras of (). It is shown that and are C* independent if and only if for all natural numbers n, m, for all n-tuples A = (A₁, ..., Aₙ) of doubly commuting nonzero operators of and m-tuples B = (B₁, ..., Bₘ) of doubly commuting nonzero operators of , $S p (A, B) = S p (A) \times S p (B)$ , where Sp denotes the joint Taylor spectrum.

The essential spectrum of holomorphic Toeplitz operators on $H^{p}$ spaces

Mats Andersson, Sebastian Sandberg (2003)

Studia Mathematica

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We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators $T_{g}$ on $H^{p} (D)$ , where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of $T_{g}$ provided that $g^{- 1} (0)$ is a compact subset of D.

Spectrum of L

W. Marek, K. Rasmussen

Similarity:

CONTENTS0. Motivation, results to be used in the sequel ................51. Slicing $L_{α}$ ’s ..........................................................102. Hereditarily countable, definable elements ................133. Spectrum of L.............................................................154. The width of elements of spectrum ............................195. Non-uniform strong definability ..................................266. Solution to a problem of Wilmers................................327....

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

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For a linear operator T in a Banach space let $σ_{p} (T)$ denote the point spectrum of T, let $σ_{p, n} (T)$ for finite n > 0 be the set of all $λ \in σ_{p} (T)$ such that dim ker(T - λ) = n and let $σ_{p, \infty} (T)$ be the set of all $λ \in σ_{p} (T)$ for which ker(T - λ) is infinite-dimensional. It is shown that $σ_{p} (T)$ is $ℱ_{σ}$ , $σ_{p, \infty} (T)$ is $ℱ_{σ δ}$ and for each finite n the set $σ_{p, n} (T)$ is the intersection of an $ℱ_{σ}$ set and a $_{δ}$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

Schur Lemma and the Spectral Mapping Formula

Antoni Wawrzyńczyk (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula $σ_{l} (S)$ = ${(λ (s))}_{s \in S} \in ℂ^{S} | {s - λ (s)}_{s \in S}$ generates a proper left ideal $.$ Using the Schur lemma and the Gelfand-Mazur theorem we prove that $σ_{l} (S)$ has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.

Rosenthal operator spaces

M. Junge, N. J. Nielsen, T. Oikhberg (2008)

Studia Mathematica

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In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an $L_{p}$ -space, then it is either an $L_{p}$ -space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non-commutative $L_{p}$ -spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which...

Geometry of Banach spaces and biorthogonal systems

S. Dilworth, Maria Girardi, W. Johnson (2000)

Studia Mathematica

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A separable Banach space X contains $ℓ_{1}$ isomorphically if and only if X has a bounded fundamental total $w c_{0} *$ -stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total $w c_{0} *$ -biorthogonal system.

Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee, Quoc-Phong Vu (2014)

Studia Mathematica

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We consider simultaneous solutions of operator Sylvester equations $A_{i} X - X B_{i} = C_{i}$ (1 ≤ i ≤ k), where $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C ₁, . . ., C_{k})$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

Noncommutative uniform algebras

Mati Abel, Krzysztof Jarosz (2004)

Studia Mathematica

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We show that a real Banach algebra A such that ||a²|| = ||a||² for a ∈ A is a subalgebra of the algebra $C_{ℍ} (X)$ of continuous quaternion-valued functions on a compact set X.

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

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We give a full solution of the following problems concerning the spaces $M_{φ} (X ⃗)$ : (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{φ} (X ⃗) \neq M_{ψ} (X ⃗)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{φ} (X ⃗) ⊊ M_{ψ} (X ⃗)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{φ} (X ⃗)$ -space for some (unknown beforehand) Banach couple X⃗.

Shift-invariant functionals on Banach sequence spaces

Albrecht Pietsch (2013)

Studia Mathematica

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The present paper is a continuation of [23], from which we know that the theory of traces on the Marcinkiewicz operator ideal $(H) : = T \in (H) : s u p_{1 \leq m < \infty} 1 / (l o g m + 1) \sum_{n = 1}^{m} a ₙ (T) < \infty$ can be reduced to the theory of shift-invariant functionals on the Banach sequence space $(ℕ ₀) : = c = (γ_{l}) : s u p_{0 \leq k < \infty} 1 / (k + 1) \sum_{l = 0}^{k} | γ_{l} | < \infty$ . The final purpose of my studies, which will be finished in [24], is the following. Using the density character as a measure, I want to determine the size of some subspaces of the dual *(H). Of particular interest are the sets formed by the Dixmier traces and the Connes-Dixmier...

On (Co)homology of triangular Banach algebras

Mohammad Sal Moslehian (2005)

Banach Center Publications

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Suppose that A and B are unital Banach algebras with units $1_{A}$ and $1_{B}$ , respectively, M is a unital Banach A,B-module, $= [\begin{matrix} A & M \\ 0 & B \end{matrix}]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{A A} = 1_{A} X 1_{A}$ , $X_{B B} = 1_{B} X 1_{B}$ , $X_{A B} = 1_{A} X 1_{B}$ and $X_{B A} = 1_{B} X 1_{A}$ . Applying two nice long exact sequences related to A, B, , X, $X_{A A}$ , $X_{B B}$ , $X_{A B}$ and $X_{B A}$ we establish some results on (co)homology of triangular Banach algebras.

Non-regularity for Banach function algebras

J. Feinstein, D. Somerset (2000)

Studia Mathematica

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Let A be a unital Banach function algebra with character space $Φ_{A}$ . For $x \in Φ_{A}$ , let $M_{x}$ and $J_{x}$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_{x}$ is connected, and that if x does not belong to the Shilov boundary of A then the set $y \in Φ_{A} : M_{x} \supseteq J_{y}$ has an infinite connected subset. Various related results are given.

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$ , belonging to the class $L^{p}(J,E)$ , $p>1$ . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

$H^{\infty}$ functional calculus in real interpolation spaces, II

Giovanni Dore (2001)

Studia Mathematica

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Let A be a linear closed one-to-one operator in a complex Banach space X, having dense domain and dense range. If A is of type ω (i.e.the spectrum of A is contained in a sector of angle 2ω, symmetric about the real positive axis, and ${| | λ (λ I - A)}^{- 1} | |$ is bounded outside every larger sector), then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the intersection of the domain and the range of the operator itself.