Spectral picture of contractions
by
László Kérchy
University of Szeged, Hungary

Let T be a Hilbert space contraction of class C₁₀, that is for every non-zero vector x we have lim_n→∞∥Tⁿ x∥ > 0 and lim_n→∞∥T^*nx∥=0. A unitary asymptote (X, W) can be associated with T, which is uniquely determined up to isomorphism. We recall that W is a unitary operator, the contractive mapping X intertwines T with W:  XT=WX, and for every unitary W' and for every intertwining X':  TX'=W'X', there exists a unique Y such that YW=W'Y,   X'=YX and ∥Y∥ ≤ ∥X'∥. It is known that the spectrum of the absolutely continuous unitary operator W is neatly contained in the spectrum of T: s(W) ⊂ s(T) and s(W)∩s' is of positive Lebesgue measure, whenever s' is a non-empty closed-and-open subset of s(T). By a former result of ours these are the only constrains on the spectra of T and W. We show that the residual set r_T of T, which carries the spectral measure of W, is also arbitrary under the condition that its essential closure is s(W). Furthermore, we prove that the general spectral picture is exhibited in the cyclic case too. (Note that the invariant subspace problem is open for C₁₀-contractions.)

Date received: May 30, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawh-68.