Topologically Invertible Elements and Topological Spectrum

Mati Abel; Wiesław Żelazko

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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....

Fermi Golden Rule, Feshbach Method and embedded point spectrum

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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.

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O. Bel Hadj Fredj, M. Burgos, M. Oudghiri (2008)

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We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space X has finite dimension if and only if the essential ascent of every operator on X is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some finite rank power if and only if $σ_{a s c}^{e} (T + F) = σ_{a s c}^{e} (T)$ for every operator T commuting with F. The quasi-nilpotent part, the analytic core and the single-valued...

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Let A be a commutative Banach algebra and let $Σ_{A}$ be its structure space. The norm spectrum σ(f) of the functional f ∈ A* is defined by $σ (f) = \bar{f \cdot a : a \in A} \cap Σ_{A}$ , where f·a is the functional on A defined by ⟨f·a,b⟩ = ⟨f,ab⟩, b ∈ A. We investigate basic properties of the norm spectrum in certain classes of commutative Banach algebras and present some applications.

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Given Banach algebras A and B with spectrum σ(B) ≠ ∅, and given θ ∈ σ(B), we define a product $A \times_{θ} B$ , which is a strongly splitting Banach algebra extension of B by A. We obtain characterizations of bounded approximate identities, spectrum, topological center, minimal idempotents, and study the ideal structure of these products. By assuming B to be a Banach algebra in ₀(X) whose spectrum can be identified with X, we apply our results to harmonic analysis, and study the question of spectral...

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Let $T$ be an operator acting on a Banach space $X$ , let $σ (T)$ and $σ_{B W} (T)$ be respectively the spectrum and the B-Weyl spectrum of $T$ . We say that $T$ satisfies the generalized Weyl’s theorem if $σ_{B W} (T) = σ (T) ∖ E (T)$ , where $E (T)$ is the set of all isolated eigenvalues of $T$ . The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$ , extending an earlier result of J. K. Finch...

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Let be a complex Banach space and e ∈ a nonzero vector. Then the set of all operators T ∈ ℒ() with $σ_{T} (e) = σ_{δ} (T)$ , respectively $r_{T} (e) = r (T)$ , is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.

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Let $T$ be a bounded representation of a commutative Banach algebra $B$ . The following spectral sets are studied. $Λ_{1} (T)$ : the Gelfand space of the quotient algebra $B / Ker T$ . $Λ_{2} (T)$ : the Gelfand space of the operator algebra $Im T$ . $Λ_{3} (T)$ : those characters $φ$ of $B$ for which the inequalities $∥ T_{b} x - \hat{b} (φ) x ∥ < ϵ ∥ x ∥$ , $b \in F$ , have a common solution $x \neq 0$ , for any $ϵ > 0$ and any finite subset $F$ of $B$ . A theorem of Beurling on the spectrum of $L^{\infty}$ -functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of...

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Qingping Zeng, Qiaofen Jiang, Huaijie Zhong (2013)

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Burgos, Kaidi, Mbekhta and Oudghiri [J. Operator Theory 56 (2006)] provided an affirmative answer to a question of Kaashoek and Lay and proved that an operator F is of power finite rank if and only if $σ_{d s c} (T + F) = σ_{d s c} (T)$ for every operator T commuting with F. Later, several authors extended this result to the essential descent spectrum, left Drazin spectrum and left essential Drazin spectrum. In this paper, using the theory of operators with eventual topological uniform descent and the technique used...

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Algebra isomorphisms between standard operator algebras

Thomas Tonev, Aaron Luttman (2009)

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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...