Displaying similar documents to “On generalized property (v) for bounded linear operators”

On the perturbation functions and similarity orbits

Haïkel Skhiri (2008)

Studia Mathematica

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We show that the essential spectral radius ϱ e ( T ) of T ∈ B(H) can be calculated by the formula ϱ e ( T ) = inf · ( X T X - 1 ) : X an invertible operator, where · ( T ) is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if · ( T ) is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then d i s t ( 0 , σ e ( T ) ) = sup · ( X T X - 1 ) : X an invertible operator.

On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov (2008)

Studia Mathematica

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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator V ϕ : f ( x ) 0 ϕ ( x ) f ( t ) d t be defined on L₂[0,1]. We prove that V ϕ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator V ϕ always equals 1.

A-Browder-type theorems for direct sums of operators

Mohammed Berkani, Mustapha Sarih, Hassan Zariouh (2016)

Mathematica Bohemica

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We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties ( SBaw ) , ( SBab ) , ( SBw ) and ( SBb ) are not preserved under direct sums of operators. However, we prove that if S and T are bounded linear operators acting on Banach spaces and having the property ( SBab ) , then S T has the property ( SBab ) if and only if σ SBF + - ( S T ) = σ SBF + - ( S ) σ SBF + - ( T ) , where σ SBF + - ( T ) is the upper semi-B-Weyl spectrum of T . We obtain analogous preservation results for the properties ( SBaw ) ,...

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...

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 , . . . , A k ) and ( B , . . . , B k ) are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and ( C , . . . , C k ) is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of ( A , . . . , A k ) and ( B , . . . , B k ) do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

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 σ p ( T ) such that dim ker(T - λ) = n and let σ p , ( T ) be the set of all λ σ p ( T ) for which ker(T - λ) is infinite-dimensional. It is shown that σ p ( T ) is σ , σ p , ( T ) is σ δ and for each finite n the set σ p , n ( T ) is the intersection of an 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...

On ( n , m ) - A -normal and ( n , m ) - A -quasinormal semi-Hilbertian space operators

Samir Al Mohammady, Sid Ahmed Ould Beinane, Sid Ahmed Ould Ahmed Mahmoud (2022)

Mathematica Bohemica

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The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let be a Hilbert space and let A be a positive bounded operator on . The semi-inner product h k A : = A h k , h , k , induces a semi-norm · A . This makes into a semi-Hilbertian space. An operator T A ( ) is said to be ( n , m ) - A -normal if [ T n , ( T A ) m ] : = T n ( T A ) m - ( T A ) m T n = 0 for some positive integers n and m .

Graphs with small diameter determined by their D -spectra

Ruifang Liu, Jie Xue (2018)

Czechoslovak Mathematical Journal

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Let G be a connected graph with vertex set V ( G ) = { v 1 , v 2 , ... , v n } . The distance matrix D ( G ) = ( d i j ) n × n is the matrix indexed by the vertices of G , where d i j denotes the distance between the vertices v i and v j . Suppose that λ 1 ( D ) λ 2 ( D ) λ n ( D ) are the distance spectrum of G . The graph G is said to be determined by its D -spectrum if with respect to the distance matrix D ( G ) , any graph having the same spectrum as G is isomorphic to G . We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs...

Migrativity properties of 2-uninorms over semi-t-operators

Ying Li-Jun, Qin Feng (2022)

Kybernetika

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In this paper, we analyze and characterize all solutions about α -migrativity properties of the five subclasses of 2-uninorms, i. e. C k , C k 0 , C k 1 , C 1 0 , C 0 1 , over semi-t-operators. We give the sufficient and necessary conditions that make these α -migrativity equations hold for all possible combinations of 2-uninorms over semi-t-operators. The results obtained show that for G C k , the α -migrativity of G over a semi-t-operator F μ , ν is closely related to the α -section of F μ , ν or the ordinal sum representation...

Weighted Frobenius-Perron operators and their spectra

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2017)

Mathematica Bohemica

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First, some classic properties of a weighted Frobenius-Perron operator 𝒫 ϕ u on L 1 ( Σ ) as a predual of weighted Koopman operator W = u U ϕ on L ( Σ ) will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of 𝒫 ϕ u under certain conditions.

Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk (2018)

Mathematica Bohemica

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We give a necessary and a sufficient condition for a subset S of a locally convex Waelbroeck algebra 𝒜 to have a non-void left joint spectrum σ l ( S ) . In particular, for a Lie subalgebra L 𝒜 we have σ l ( L ) if and only if [ L , L ] generates in 𝒜 a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space

Michael Gil&#039; (2024)

Czechoslovak Mathematical Journal

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Let A be a bounded linear operator in a complex separable Hilbert space , and S be a selfadjoint operator in . Assuming that A - S belongs to the Schatten-von Neumann ideal 𝒮 p ( p > 1 ) , we derive a bound for k | R λ k ( A ) - λ k ( S ) | p , where λ k ( A ) ( k = 1 , 2 , ) are the eigenvalues of A . Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case p = 2 we refine the Weyl inequality between the real parts of the eigenvalues of A and the eigenvalues...

Some results on semi-stratifiable spaces

Wei-Feng Xuan, Yan-Kui Song (2019)

Mathematica Bohemica

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We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If X is a semi-stratifiable space, then X is separable if and only if X is D C ( ω 1 ) ; (2) If X is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then X is separable; (3) Let X be a ω -monolithic star countable extent semi-stratifiable space. If t ( X ) = ω and d ( X ) ω 1 , then X is hereditarily separable. Finally, we prove that for...

The weak Gelfand-Phillips property in spaces of compact operators

Ioana Ghenciu (2017)

Commentationes Mathematicae Universitatis Carolinae

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For Banach spaces X and Y , let K w * ( X * , Y ) denote the space of all w * - w continuous compact operators from X * to Y endowed with the operator norm. A Banach space X has the w G P property if every Grothendieck subset of X is relatively weakly compact. In this paper we study Banach spaces with property w G P . We investigate whether the spaces K w * ( X * , Y ) and X ϵ Y have the w G P property, when X and Y have the w G P property.

Recurrence and mixing recurrence of multiplication operators

Mohamed Amouch, Hamza Lakrimi (2024)

Mathematica Bohemica

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Let X be a Banach space, ( X ) the algebra of bounded linear operators on X and ( J , · J ) an admissible Banach ideal of ( X ) . For T ( X ) , let L J , T and R J , T ( J ) denote the left and right multiplication defined by L J , T ( A ) = T A and R J , T ( A ) = A T , respectively. In this paper, we study the transmission of some concepts related to recurrent operators between T ( X ) , and their elementary operators L J , T and R J , T . In particular, we give necessary and sufficient conditions for L J , T and R J , T to be sequentially recurrent. Furthermore, we prove that L J , T is recurrent...