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The paper deals with covering problems and the degree of compactness of operators. The main part is devoted to relationships between entropy moduli and Kolmogorov (resp. Gelfand and approximation) numbers for operators which may be interpreted as counterparts to the classical Bernstein-Jackson inequalities for functions. Certain quantifications of results in the Riesz-Schauder-Theory are given. Finally, the largest distance between “the degree of approximation” and the “degree of compactness” of...
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 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.
On définit et étudie les espaces de Banach de type et de cotype , . Cela permet de dire que, dès que certaines applications sont -sommantes, elles sont aussi -sommantes pour .
This article is divided into two parts. The first one is on the linear structure of the set of norm-attaining functionals on a Banach space. We prove that every Banach space that admits an infinite-dimensional separable quotient can be equivalently renormed so that the set of norm-attaining functionals contains an infinite-dimensional vector subspace. This partially solves a question proposed by Aron and Gurariy. The second part is on the linear structure of dominated operators. We show that the...
2000 Mathematics Subject Classification: 46B28, 47D15.In this paper we introduce and study the lp-lattice summing operators in the category of operator spaces which are the analogous of p-lattice summing operators in the commutative case. We study some interesting characterizations of this type of operators which generalize the results of Nielsen and Szulga and we show that Λ l∞( B(H) ,OH) ≠ Λ l2( B( H) ,OH), in opposition to the commutative case.
In the paper local entropy moduli of operators between Banach spaces are introduced. They constitue a generalization of entropy numbers and moduli, and localize these notions in an appropriate way. Many results regarding entropy numbers and moduli can be carried over to local entropy moduli. We investigate relations between local entropy moduli and s-numbers, spectral properties, eigenvalues, absolutely summing operators. As applications, local entropy moduli of identical and diagonal operators...
Using elementary arguments we improve former results of P. Vrbová concerning local spectrum. As a consequence, we obtain a new proof of Kaplansky’s theorem on algebraic operators on a Banach space.
We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-valued functions , where for a fixed function ϕ, denotes the one-dimensional orthogonal projection on the function , U is a group representation and g is an element of the group. They are defined as integrals , where W is an open, relatively...
Concise but self-contained reviews are given on theories of majorization and symmetrically normed ideals, including the proofs of the Lidskii-Wielandt and the Gelfand-Naimark theorems. Based on these reviews, we discuss logarithmic majorizations and norm inequalities of Golden-Thompson type and its complementary type for exponential operators on a Hilbert space. Furthermore, we obtain norm convergences for the exponential product formula as well as for that involving operator means.
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