Compact operators between real interpolation spaces

{orders}ms/98424/cobos.3d -17.11.00 - 11:40 REAL INTERPOLATION OF COMPACT OPERATORS BETWEEN QUASI-BANACH SPACES

Let …A0;A1† and …B0;B1† be couples of quasi-Banach spaces and let T be a linear operator. We prove that if T : A0 ! B0 is compact and T : A1 ! B1 is bounded, then T : …A0;A1† ;q ! …B0;B1† ;q is also compact. Some results on the structure of minimal and maximal interpolation methods are also established. 0. Introduction. Assume that T is a linear operator such that T : Lp0 ! Lq0 compactly and T ...

Compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions

We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.

Compact Weighted Composition Operators and Multiplication Operators between Hardy Spaces

Complex Interpolation of Compact Operators

If (A0, A1) and (B0, B1) are Banach couples and a linear operator T : A0 +A1 → B0 +B1 maps A0 compactly into B0 and maps A1 boundedly into B1, does T necessarily also map [A0, A1]θ compactly into [B0, B1]θ for θ ∈ (0, 1)? After 42 years this question is still not answered, not even in the case where T : A1 → B1 is also compact. But affirmative answers are known for many special choices of (A0, ...

REAL INTERPOLATION OF DOMAINS OF SECTORIAL OPERATORS ON Lp-SPACES

Let A be a sectorial operator on a non-atomic Lp-space, 1 ≤ p < ∞, whose resolvent consists of integral operators, or more generally, has a diffuse representation. Then the fractional domain spaces D(Aα) for α ∈ (0, 1) do not coincide with the real interpolation spaces of (Lq , D(A)). As a consequence, we obtain that no such operator A has a bounded H∞-calculus if p = 1.