Measure of weak noncompactness under complex interpolation
Volume 147 / 2001
Studia Mathematica 147 (2001), 89-102
MSC: 46B70, 46M35.
DOI: 10.4064/sm147-1-7
Abstract
Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón's complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if $T:A_{0}\rightarrow B_{0}$ or $T:A_{1}\rightarrow B_{1}$ is weakly compact, then so is $T:A_{[\theta ]}\rightarrow B_{[\theta ]}$ for all $0<\theta <1$, where $A_{[\theta ]}$ and $B_{[\theta ]}$ are interpolation spaces with respect to the pairs $(A_{0},A_{1})$ and $(B_{0},B_{1})$. Some formulae for this measure and relations to other quantities measuring weak noncompactness are established.
