Bernd Carl

Entropy numbers, s-numbers, and eigenvalue problems

Abstract We establish inequalities between entropy numbers and approximation numbers for operators acting between Banach spaces. Furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a Banach space. The results are compared with the classical inequalities of Bernstein and Jackson.

Gelfand numbers of operators with values in a Hilbert space

SummaryIn the paper we prove two inequalities involving Gelfand numbers of operators with values in a Hilbert space. The first inequality is a Rademacher version of the main result in [Pa-To-1] which relates the Gelfand numbers of an operator from a Banach spaceX intol2n with a certain Rademacher average for the dual operator. The second inequality states that the Gelfand numbers of an operatoru froml1N into a Hilbert space satisfy the inequality

s-Numbers of integral operators with Hölder continuous kernels over metric compacta

k^{1/2} c_k (u) \leqq C\parallel u\parallel (\log (1 + N/k))^{1/2}

Tensor products and Grothendieck type inequalities of operators in _{}-spaces

whereC is a universal constant. Several applications of these inequalities in the geometry of Banach spaces are given.

Entropy numbers of convex hulls in Banach spaces and applications

In this paper we characterize diagonal operators from ZP into l,, 1 <p, q < co, by their entropy numbers. The results contain those of Marcus [ 161 formulated in the language of c-entropy and Oloff [21]. Moreover, the remaining gaps in [ 16, 2 I] are tilled in the much more complicated situation where l<p<q<co. Furthermore, we extend the results to diagonal operators acting between Lorentz sequence spaces. It turns out that the computation of entropy numbers of diagonal operators can be reduced to the computation of certain entropy quasi-ideal norms of identity operators acting between the simple itdimensional vector spaces I;. Finally, the entropy numbers are used for studying eigenvalue problems of factorable operators acting on Banach spaces. The statements of this paper are obtained using results and techniques recently proved and developed by the author in [4].

On a Weyl inequality of operators in Banach spaces

Abstract We determine the asymptotic behaviour of Kolmogorov and approximation numbers of integral operators, which act in C ( X ), where X is a compact metric space, and which are defined by means of Holder continuous kernels. The results show the connection of the decay of these s -numbers with the entropy (and in this way with the dimension) of X .

An inequality between thep- and (p, 1)-summing norm of finite rank operators fromC(K)-spaces

Abstract Let S be an operator admitting a factorization where D is a diagonal operator and T an (arbitrary) operator with the image in a Banach space of type p . We shall characterize these operators by entropy numbers. Using a well-known inequality between eigenvalues and entropy numbers we establish some results about distributions of eigenvalues.

Local Entropy Moduli and Eigenvalues of Operators in Banach Spaces.

Several results in the theory of (p, q)-summing operators are improved by a unified but elementary tensor product concept