We associate to any contraction T (and, more generally, to any operator T of class C \ ) in a von Neumann algebra M an operator kernel K : (T) (|:| <1) which allows us to define various kinds of functional calculis for T. When M is finite, we use this kernel to give a short proof of the Fuglede Kadison theorem on the location of the trace and to prove that a contraction T in M is unitary if and only if its spectrum is contained in the unit circle. By using a perturbation of the kernel K : (T) we give, for any operator T of class C \ acting on a separable Hilbert space H, a short proof of the power inequality for the numerical range and an accurate conjugacy (to a contraction) result for T. We also get a generalized von Neumann inequality which gives a good control of & f (rT*) x+g(rT ) x& (0 r<1) for x # H and f, g in the disc algebra. Finally, we associate to any C 1 contraction in a Hilbert space an asymptotic kernel which allows us to describe new kinds of invariant subspaces for T, from the positive solutions X of the operator equation T*XT=X. In particular, we recover some results of Beauzamy based on the notion of invariant subspace of ``functional type. '' 1996 Academic Press, Inc. 0. Preliminaries 0.1. Introduction.
The starting point of this paper is a question raised by P. de la Harpe, Robertson, and Valette: ``Can one hear the shape of a cyclic group among all discrete finitely generated groups?'', which easily reduces to know article no.