Md-numbers and Asymptotic Md-numbers of Operators

First, the basic properties of mean dilatation (MD-) numbers for linear operators acting from a finite-dimensional Hilbert space are investigated. Among other results, in terms of first and second order MD-numbers, a characterization of isometries is obtained and a dimension-free estimation of the p-th order MD-number by means of the first order MD-number is established. After that asymptotic MD-numbers for a continuous linear operator acting from an infinite-dimensional Hilbert space are introduced and it is shown that in the case of an infinite-dimensional domain the asymptotic p-th order MD-number, rather unexpectedly, is simply the p-th power of the asymptotic first order MD-number (Theorem 3.1). 2000 Mathematics Subject Classification: Primary: 46C05; Secondary: 47A30, 47B06.