Carsten Michels

A complex interpolation formula for tensor products of vector-valued Banach function spaces

Abstract. We prove the complex interpolation formula

Summing inclusion maps between symmetric sequence spaces

[{X_0}{(E_0)} \tilde{\otimes} _{\varepsilon} {Y_0}{(F_0)}, {X_1}{(E_1)} \tilde {\otimes} _{\varepsilon } {Y_1}{(F_1)}]_\theta = [{X_0}{(E_0)},{X_1}{(E_1)}]_\theta \tilde {\otimes} _{\varepsilon } [{Y_0}{(F_0)},{Y_1}{(F_1)}]_\theta ,

Summing inclusion maps between symmetric sequence spaces, a survey

for the injective tensor product of vector-valued Banach function spaces Xi(Ei) and Yi(Fi) satisfying certain geometric assumptions. This result unifies results of Kouba, and moreover, our approach offers an alternate proof of Koubas interpolation formula for scalar-valued Banach function spaces.

Orlicz norm estimates for eigenvalues of matrices

In 1973/74 Bennett and (independently) Carl proved that for 1 < u < 2 the identity map id: l u → l 2 is absolutely (u, 1)-summing, i.e., for every unconditionally summable sequence (x n ) in l u the scalar sequence (∥x n ∥l 2 ) is contained in l u , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a 2-concave symmetric Banach sequence space E the identity map id: E → l 2 is absolutely (E,1)-summing, i.e., for every unconditionally summable sequence (x n ) in E the scalar sequence (∥x n ∥ l2 ) is contained in E. Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator T on l 2 with values in a 2-concave symmetric Banach sequence space E is a multiplier from l 2 into E. Furthermore, we prove an asymptotic formula for the k-th approximation number of the identity map id: l n 2 → E n , where E n denotes the linear span of the first n standard unit vectors in E, and apply it to Lorentz and Orlicz sequence spaces.

Eigenvalue estimates for operators on symmetric Banach sequence spaces

Abstract For 1 ≤ p ≤ 2 let E be a p-concave symmetric Banach sequence space, so in particular contained in l p . It is proved in [14] and [15] that for each weakly 2 -summable sequence ( x n ) in E the sequence (║ x n ║ p ) of norms in l p is a multiplier from l p into E. This result is a proper improvement of well-known analogues in l p -spaces due to Littlewood, Orlicz, Bennett and Carl, which had important impact on various parts of analysis. We survey on a series of recent articles around this cycle of ideas, and prove new results on approximation numbers and strictly singular operators in sequence spaces. We also give applications to the theories of eigenvalue distribution and interpolation of operators.

Summing norms of identities between unitary ideals

AbstractLet φ be a supermultiplicative Orlicz function such that the function

Bennett-Carl inequalities for symmetric Banach sequence spaces and unitary ideals

t \mapsto \varphi \left( {\sqrt t } \right)

Complex interpolation of spaces of operators on l1

is equivalent to a convex function. Then each complexn×n matrixT=(τij)i, j satisfies the following eigenvalue estimate: