Measures on spaces of operators and isometries

Abstract

One considers a question that arises at the investigation of isometric operators in vector-valued L^p -spaces. Let E, F be Banach spaces, let p>0, let μ be a probability Borel measure on the space on continuous linear operators from E into F such that for any e ε E one has

$$\left\| e \right\|^p = \int {\left\| {Te} \right\|^p } d\mu \left( T \right)$$

. In the cases when: 1) E=C(K), K is a metric compactum, F is an arbitrary space, p>1 and 2) 2)E=F=L^q,p>1, q>1 q ∉[p,2] it is Proved that the support of the measure μ is contained in the set of the operators that are scalar multiples of isometries. For E=C(K) one obtains an isomorphic analogue of this result: if the Banach-Mazur distance between C(K) and the P -sum of Banach spaces is small, then the distance between C(K) and one of the spaces is small.