On operators with the same local spectra

Aleksandar Torgašev

Displaying similar documents to “On operators with the same local spectra”

On the norm-closure of the class of hypercyclic operators

Christoph Schmoeger (1997)

Annales Polonici Mathematici

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Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if $f (σ_{W} (T)) \cup z \in ℂ : | z | = 1$ is connected, where $σ_{W} (T)$ denotes the Weyl spectrum of T.

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

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Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X, | | \cdot | |_{X})$ and $(Y, | | \cdot | |_{Y})$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $| | T (1_{A ₙ} f) {| |}_{Y} \to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

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.

Multiple summing operators on $l_{p}$ spaces

Dumitru Popa (2014)

Studia Mathematica

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We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T : l_{4 / 3} \times l_{4 / 3} \to l ₂$ such that none of the associated linear operators...

Regularity of domains of parameterized families of closed linear operators

Teresa Winiarska, Tadeusz Winiarski (2003)

Annales Polonici Mathematici

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The purpose of this paper is to provide a method of reduction of some problems concerning families $A_{t} = {(A (t))}_{t \in}$ of linear operators with domains ${(_{t})}_{t \in}$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${(Ψ_{t})}_{t \in}$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t \mapsto Ψ_{t}$ is sufficiently regular and (ii) $Ψ_{t} () =_{t}$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition...

A new characterization of Anderson’s inequality in $C_{1}$ -classes

S. Mecheri (2007)

Czechoslovak Mathematical Journal

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Let $ℋ$ be a separable infinite dimensional complex Hilbert space, and let $ℒ (ℋ)$ denote the algebra of all bounded linear operators on $ℋ$ into itself. Let $A = (A_{1}, A_{2}, \dots, A_{n})$ , $B = (B_{1}, B_{2}, \dots, B_{n})$ be $n$ -tuples of operators in $ℒ (ℋ)$ ; we define the elementary operators $Δ_{A, B} ℒ (ℋ) \mapsto ℒ (ℋ)$ by $Δ_{A, B} (X) = \sum_{i = 1}^{n} A_{i} X B_{i} - X .$ In this paper, we characterize the class of pairs of operators $A, B \in ℒ (ℋ)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A, B \in ℒ (ℋ)$ such that $\sum_{i = 1}^{n} B_{i} T A_{i} = T$ implies $\sum_{i = 1}^{n} A_{i}^{*} T B_{i}^{*} = T$ for all $T \in 𝒞_{1} (ℋ)$ (trace class operators). The main result is the equivalence between this property and the...

On a proof of the boundedness and nuclearity of pseudodifferential operators in $R^{n}$

Jan A. Rempała (1990)

Annales Polonici Mathematici

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Besov spaces and 2-summing operators

M. A. Fugarolas (2004)

Colloquium Mathematicae

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Let Π₂ be the operator ideal of all absolutely 2-summing operators and let $I_{m}$ be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of $I_{m}$ . Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also...

Weighted norm estimates and $L_{p}$ -spectral independence of linear operators

Peer C. Kunstmann, Hendrik Vogt (2007)

Colloquium Mathematicae

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We investigate the $L_{p}$ -spectrum of linear operators defined consistently on $L_{p} (Ω)$ for p₀ ≤ p ≤ p₁, where (Ω,μ) is an arbitrary σ-finite measure space and 1 ≤ p₀ < p₁ ≤ ∞. We prove p-independence of the $L_{p}$ -spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_{p}$ -spectral independence...

Stability of infinite ranges and kernels

K.-H. Förster, V. Müller (2006)

Studia Mathematica

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Let A(·) be a regular function defined on a connected metric space G whose values are mutually commuting essentially Kato operators in a Banach space. Then the spaces $R^{\infty} (A (z))$ and $\bar{N^{\infty} (A (z))}$ do not depend on z ∈ G. This generalizes results of B. Aupetit and J. Zemánek.

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

Quasinilpotent operators in operator Lie algebras II

Peng Cao (2009)

Studia Mathematica

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In this paper, it is proved that the Banach algebra $\bar{(ℒ)}$ , generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and $\bar{(ℒ)}$ consists of polynomially compact operators. It is also proved that $\bar{(ℒ)}$ consists of quasinilpotent operators if ℒ is an essentially nilpotent Engel Lie algebra generated by quasinilpotent operators. Finally, Banach algebras generated by essentially nilpotent Lie algebras are shown to be compactly quasinilpotent. ...

Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

K. Frączek, M. Wysokińska (2008)

Colloquium Mathematicae

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We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on $(X ̃,_{X ̃}, m)$ . Consider the associated unitary operators on $L ² (X ̃,_{X ̃}, m)$ given by $U ̃_{S} f = \sqrt (d (m \circ S) / d m) \cdot (f \circ S)$ and $φ \cdot U ̃_{S}$ , where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

Smooth operators in the commutant of a contraction

Pascale Vitse (2003)

Studia Mathematica

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For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the $H^{\infty}$ calculus, $H^{\infty} (T)$ , and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes $S_{p}$ .

Some properties and applications of equicompact sets of operators

E. Serrano, C. Piñeiro, J. M. Delgado (2007)

Studia Mathematica

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Let X and Y be Banach spaces. A subset M of (X,Y) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xₙ) in X has a subsequence $(x_{k (n)}) ₙ$ such that $(T x_{k (n)}) ₙ$ is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness...

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.

The group of L²-isometries on H¹₀

Esteban Andruchow, Eduardo Chiumiento, Gabriel Larotonda (2013)

Studia Mathematica

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Let Ω be an open subset of ℝⁿ. Let L² = L²(Ω,dx) and H¹₀ = H¹₀(Ω) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group of invertible operators on H¹₀ which preserve the L²-inner product. When Ω is bounded and ∂Ω is smooth, this group acts as the intertwiner of the H¹₀ solutions of the non-homogeneous Helmholtz equation u - Δu = f, ${u |}_{\partial Ω} = 0$ . We show that is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable...

$^{\infty}$ -vectors and boundedness

Jan Stochel, F. H. Szafraniec (1997)

Annales Polonici Mathematici

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The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its $^{\infty}$ -vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

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.

Operators which commute with convolutions on subspaces of $L_{\infty} (G)$

Anthony To-Ming Lau (1978)

Colloquium Mathematicae

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