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Displaying similar documents to “Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space”

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.

On the range of some elementary operators

Hamza El Mouadine, Abdelkhalek Faouzi, Youssef Bouhafsi (2024)

Commentationes Mathematicae Universitatis Carolinae

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Let L ( H ) denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space H . For A , B L ( H ) , the generalized derivation δ A , B and the multiplication operator M A , B are defined on L ( H ) by δ A , B ( X ) = A X - X B and M A , B ( X ) = A X B . In this paper, we give a characterization of bounded operators A and B such that the range of M A , B is closed. We present some sufficient conditions for δ A , B to have closed range. Some related results are also given.

Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces X α , Y α and Z α such that (i) f X α , f Y α , f Z α = ω , where f is either trdef or ₀-trsur, (ii) A ( α ) - t r i n d X α = and M ( α ) - t r i n d X α = - 1 , (iii) A ( α ) - t r i n d Y α = - 1 and M ( α ) - t r i n d Y α = , and (iv) A ( α ) - t r i n d Z α = M ( α ) - t r i n d Z α = and A ( α + 1 ) M ( α + 1 ) - t r i n d Z α = - 1 . We also show that there exists no separable metrizable space W α with A ( α ) - t r i n d W α , M ( α ) - t r i n d W α and A ( α ) M ( α ) - t r i n d W α = , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

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 ) ,...

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.

More on exposed points and extremal points of convex sets in n and Hilbert space

Stoyu T. Barov (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let 𝕍 be a separable real Hilbert space, k with k < dim 𝕍 , and let B be convex and closed in 𝕍 . Let 𝒫 be a collection of linear k -subspaces of 𝕍 . A point w B is called exposed by 𝒫 if there is a P 𝒫 so that ( w + P ) B = { w } . We show that, under some natural conditions, B can be reconstituted as the convex hull of the closure of all its exposed by 𝒫 points whenever 𝒫 is dense and G δ . In addition, we discuss the question when the set of exposed by some 𝒫 points forms a G δ -set.

Linear maps preserving A -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

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Let be a complex Hilbert space, A a positive operator with closed range in ( ) and A ( ) the sub-algebra of ( ) of all A -self-adjoint operators. Assume φ : A ( ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ) = ψ ( T ) for all T A ( ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there...

A characterization of reflexive spaces of operators

Janko Bračič, Lina Oliveira (2018)

Czechoslovak Mathematical Journal

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We show that for a linear space of operators ( 1 , 2 ) the following assertions are equivalent. (i) is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = ( ψ 1 , ψ 2 ) on a bilattice Bil ( ) of subspaces determined by with P ψ 1 ( P , Q ) and Q ψ 2 ( P , Q ) for any pair ( P , Q ) Bil ( ) , and such that an operator T ( 1 , 2 ) lies in if and only if ψ 2 ( P , Q ) T ψ 1 ( P , Q ) = 0 for all ( P , Q ) Bil ( ) . This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

On the range-kernel orthogonality of elementary operators

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

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Let L ( H ) denote the algebra of operators on a complex infinite dimensional Hilbert space H . For A , B L ( H ) , the generalized derivation δ A , B and the elementary operator Δ A , B are defined by δ A , B ( X ) = A X - X B and Δ A , B ( X ) = A X B - X for all X L ( H ) . In this paper, we exhibit pairs ( A , B ) of operators such that the range-kernel orthogonality of δ A , B holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of Δ A , B with respect to the wider class of unitarily invariant...

L -limited-like properties on Banach spaces

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study weakly precompact sets and operators. We show that an operator is weakly precompact if and only if its adjoint is pseudo weakly compact. We study Banach spaces with the p - L -limited * and the p -(SR * ) properties and characterize these classes of Banach spaces in terms of p - L -limited * and p -Right * subsets. The p - L -limited * property is studied in some spaces of operators.

H calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the H calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if - A generates a bounded analytic C 0 semigroup ( T t ) on a UMD space, then the H calculus of A is bounded if and only if ( T t ) has a dilation to a bounded group on L 2 ( [ 0 , 1 ] , X ) . This generalises a Hilbert space result of C.LeMerdy. If X is an L p space we can choose another L p space in place of L 2 ( [ 0 , 1 ] , X ) .

Multiplication operators on L ( L p ) and p -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

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A classification of weakly compact multiplication operators on L ( L p ) , 1<p< , i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f p - s t r i c t l y s i n g u l a r o p e r a t o r s , a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l p - s t r i c t l y s i n g u l a r o p e r a t o r s o n Lp . T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r T o n Lp , 1<p<2 , i s p - s t r i c t l y s i n g u l a r a n d T|X i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e X o f Lp , t h e n X e m b e d s i n t o Lr f o r a l l r<2 , b u t X n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e . It is also shown that if T is convolution by a biased coin on L p of the Cantor group, 1 p < 2 , and T | X is an isomorphism for some reflexive subspace X of L p , then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976.