Fernando Cobos

One-sided compactness results for Aronszajn-Gagliardo functors

Interpolating compactness properties of operators is a long standing and important problem. In this paper, the authors consider the problem in a very general setting of Aronszajn-Gagliardo functors. In simplest terms they show that if T : A0 ! B0 is compact and T : A1 ! B1 is bounded, then T is compact on some interpolation spaces constructed in the Aronszajn-Gagliardo methods. These methods do not include the complex interpolation method, but the authors show that if the two couples are formed by Banach lattices then the theorem holds in the complex method as well. Additional results are in the context of interpolation of compactness properties in the context of N-tuples of Banach spaces. The paper includes a number of related, interesting results.

Interpolation of compactness using Aronszajn-Gagliardo functors

AbstractWe prove that ifT: A0 →B0 andT: A1 →B1 both are compact, thenn

On a limit class of approximation spaces

T:F(bar A) to F(bar B)

Eigenvalues of integral operators with positive definite kernels satisfying integrated Ho¨lder conditions over metric compacta

n is also compact, whereF is the minimal or the maximal functor in the sense of Aronszajn-Gagliardo. We also derive some results for ordered couples.

Entropy and eigenvalues of weakly singular integral operators

We prove that if T: A0 → B0 is compact and T: A1 → B1 is compact (or T: A1 → B1 is bounded and, then given any θ and q with 0 < θ < 1 and 0 < q⩽ ∞, it follows that T: (A0, A1)0,q → (B0, B1)0,q, is also compact. Here (A0, A1)0,q and (B0, B1)0,q are the usual real interpolation spaces.

Entropy and Lorentz-Marcinkiewicz operator ideals

We describe the behaviour under interpolation of a limit class of approximation spaces. We characterize them as extrapolation spaces. Moreover, we study the boundedness of certain operators on these spaces. As an application, we derive several results on Macaev operator ideals.

On Hille-Tamarkin operators and Schatten classes

The authors extend a result of K. Hayakawa [J. Math. Soc. Jap. 21, 189-199 (1969; Zbl 0181.137)], and prove: If T is a linear operator such that T: A0 ! B0, is bounded,and T: A1 ! B1 is compact, and moreover, A1 A0, then T: ¯ A,q ! ¯B,q is compact for 0 < < 1, 0 < q 1.

Geometric Aspects of Banach Spaces: On some operator ideals defined by approximation numbers

Abstract We determine the asymptotic eigenvalue behaviour of integral operators generated by positive definite kernels satisfying an integrated Holder condition on metric compacta. We also show that this behaviour is the best possible.