Given any operator ideal , there are two natural functionals , that one can use to show the deviation of the operator to the closed surjective hull of and to the closed injective hull of , respectively. We describe the behaviour under interpolation of and . The results are part of joint works with A. Martínez, A. Manzano and P. Fernández-Martínez.
The paper is devoted to some aspects of the real interpolation method in the case of triples (X₀,X₁,Q) where X̅: = (X₀,X₁) is a Banach couple and Q is a convex cone. The first fundamental result of the theory, the interpolation theorem, holds in this situation (for linear operators preserving the cone structure). The second one, the reiteration theorem, holds only under some conditions on the triple. One of these conditions, the so-called intersection property, is studied for cones with respect...
We prove that the basic facts of the real interpolation method remain true for couples of cones obtained by intersection of the cone of concave functions with rearrangement invariant spaces.
We study the connection between intersection properties of balls and the existence of large faces of the unit ball in Banach spaces. Hanner’s result that a real space has the 3.2 intersection property if an only if disjoint faces of the unit ball are contained in parallel hyperplanes is extended to infinite dimensional spaces. It is shown that the space of compact operators from a space to a space has the 3.2 intersection property if and only if and have the 3.2 intersection property and...
A space is called -compact by M. Mandelker if the intersection of all free maximal ideals of coincides with the ring of all functions in with compact support. In this paper we introduce -compact and -compact spaces and we show that a space is -compact if and only if it is both -compact and -compact. We also establish that every space admits a -compactification and a -compactification. Examples and counterexamples are given.
Suppose that X is a Banach space of analytic functions on a plane domain Ω. We characterize the operators T that intertwine with the multiplication operators acting on X.
We give characterizations of Besov and Triebel-Lizorkin spaces and in smooth domains via convolutions with compactly supported smooth kernels satisfying some moment conditions. The results for s ∈ ℝ, 0 < p,q ≤ ∞ are stated in terms of the mixed norm of a certain maximal function of a distribution. For s ∈ ℝ, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ characterizations without use of maximal functions are also obtained.
We present a way of thinking of exponential farnilies as geodesic surfaces in the class of positive functions considered as a (multiplicative) sub-group G+ of the group G of all invertible elements in the algebra A of all complex bounded functions defined on a measurable space. For that we have to study a natural geometry on that algebra. The class D of densities with respect to a given rneasure will happen to be representatives of equivalence classes defining a projective space in A. The natural...