Contractions in von Neumann algebras

We associate to any contraction T (and, more generally, to any operator T of class C-p) in a von Neumann algebra M an operator kernel K-alpha (T) (/alpha/ <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-alpha(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 parallel to f(rT*)x+g(rT)x parallel to (0 less than or equal to r 1) for x is an element of 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.'' (C) 1996 Academic Press, Inc.