Weak-type interpolation and orbits

Orbits, Weak Orbits and Local Capacity of Operators

Let T be an operator on a Banach space X. We give a survey of results concerning orbits {Tnx : n = 0, 1, . . .} and weak orbits {〈Tnx, x∗〉 : n = 0, 1, . . .} of T where x ∈ X and x∗ ∈ X∗. Further we study the local capacity of operators and prove that there is a residual set of points x ∈ X with the property that the local capacity cap(T, x) is equal to the global capacity cap T . This is an an...

On weak orbits of operators

Let T be a completely nonunitary contraction on a Hilbert space H with r(T ) = 1. Let an > 0, an → 0. Then there exists x ∈ H with |〈Tnx, x〉| ≥ an for all n. We construct a unitary operator without this property. This gives a negative answer to a problem of van Neerven. Let X be a complex Banach space. Then each operator T ∈ B(X) has orbits that are ”large” in the following sense [M1], [B]: Let...

Interpolation orbits in the Lebesgue spaces

This paper is devoted to description of interpolation orbits with respect to linear operators mapping an arbitrary couple of L p spaces with weights {L p 0 (U 0), L p 1 (U 1)} into an arbitrary couple {L q 0 (V 0), L q 1 (V 1)}, where 1 ≤ p 0 , p 1 , q 0 , q 1 ≤ ∞. By L p (U) we denote the space of measurable functions f on a measure space M such that f U ∈ L p with the norm f Lp(U) = f U Lp. }...

Interpolation and Almost Interpolation by Weak Chebyshev Spaces

Some new results on univariate interpolation by weak Cheby-shev spaces, using conditions of Schoenberg-Whitney type and the concept of almost interpolation sets, are given. x1. Introduction Let U denote a nite-dimensional subspace of real-valued functions deened on some set K. We are interested in describing those conngurations T = dimU jT = s: T is called an interpolation set (I-set) w.r.t. U....

Weak Interpolation in Banach Spaces