Displaying similar documents to “A remark on the range of elementary operators”

Isolated points of spectrum of k-quasi-*-class A operators

Salah Mecheri (2012)

Studia Mathematica

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Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class, denoted *, of operators satisfying T * k ( | T ² | - | T * | ² ) T k ≥ 0 where k is a natural number, and we prove basic structural properties of these operators. Using these results, we also show that if E is the Riesz idempotent for a non-zero isolated point μ of the spectrum of T ∈ *, then E is self-adjoint and EH = ker(T-μ) = ker(T-μ)*. Some spectral properties are also presented.

Some remarks on quasi-Cohen sets

Pascal Lefèvre, Daniel Li (2001)

Colloquium Mathematicae

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We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection C ( G ) / C E c ( G ) ↪ L ² E ( G ) is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.

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 , ⋯ , A n ) , B = ( B 1 , B 2 , ⋯ , B n ) be n -tuples of operators in ℒ ( ℋ ) ; we define the elementary operators Δ A , B ℒ ( ℋ ) ↦ ℒ ( ℋ ) by Δ A , B ( X ) = ∑ i = 1 n A i X B i - X . In this paper, we characterize the class of pairs of operators A , B ∈ ℒ ( ℋ ) satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators A , B ∈ ℒ ( ℋ ) such that ∑ i = 1 n B i T A i = T implies ∑ i = 1 n A i * T B i * = T for all T ∈ 𝒞 1 ( ℋ ) (trace class operators). The main result is the equivalence between this property and the...

Quasi-constricted linear operators on Banach spaces

Eduard Yu. Emel'yanov, Manfred P. H. Wolff (2001)

Studia Mathematica

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Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace X ₀ : = x ∈ X : l i m n → ∞ | | T ⁿ x | | = 0 is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness χ | | · | | ₁ ( A ) < 1 for some equivalent norm ||·||₁ on X. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator T by quasi-constrictedness of scalar multiples of T. Finally, we prove...

On L w 2 -quasi-derivatives for solutions of perturbed general quasi-differential equations

Sobhy El-sayed Ibrahim (1999)

Czechoslovak Mathematical Journal

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This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of n th order with complex coefficients M [ y ] - λ w y = w f ( t , y [ 0 ] , ... , y [ n - 1 ] ) , t ∈ [ a , b ) provided that all r th quasi-derivatives of solutions of M [ y ] - λ w y = 0 and all solutions of its normal adjoint M + [ z ] - λ ¯ w z = 0 are in L w 2 ( a , b ) and under suitable conditions on the function f .

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...

On quasi-compactness of operator nets on Banach spaces

Eduard Yu. Emel&#039;yanov (2011)

Studia Mathematica

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The paper introduces a notion of quasi-compact operator net on a Banach space. It is proved that quasi-compactness of a uniform Lotz-Räbiger net ( T λ ) λ is equivalent to quasi-compactness of some operator T λ . We prove that strong convergence of a quasi-compact uniform Lotz-Räbiger net implies uniform convergence to a finite-rank projection. Precompactness of operator nets is also investigated.

On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

Sameer Chavan (2011)

Studia Mathematica

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The Cauchy dual operator T’, given by T ( T * T ) - 1 , provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1...

Fine and quasi connectedness in nonlinear potential theory

David R. Adams, John L. Lewis (1985)

Annales de l'institut Fourier

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If B α , p denotes the Bessel capacity of subsets of Euclidean n -space, α &gt; 0 , 1 &lt; p &lt; ∞ , naturally associated with the space of Bessel potentials of L p -functions, then our principal result is the estimate: for 1 &lt; α p ≤ n , there is a constant C = C ( α , p , n ) such that for any set E min { B α , p ( E ∩ Q ) , B α , p ( E c ∩ Q ) } ≤ C · B α , p ( Q ∩ ∂ f E ) for all open cubes Q in n -space. Here ∂ f E is the boundary of the E in the ( α , p ) -fine topology i.e. the smallest topology on c -space that makes the associated ( α , p ) -linear potentials continuous there. As a consequence,...