IMB, Université de Bordeaux 1
Date(s) : 17/11/2014 iCal
10 h 00 min - 11 h 00 min
On a natural generalization of rho-contractions
A result of Eckstein asserts that for any ρ-contraction T on a Hilbert space H the sequence (∥Tnh∥)n is convergent for any h ϵ H. We show that this remains true for a natural generalization of the class of ρ-contractions, which we call the class of (ρ,N)-contractions (notation: Cρ,N(H)). Our argument follows the lines of Mlak’s proof of Eckstein’s result, but is somewhat simplified by a study of coisometric (ρ,N)-dilations of these operators, which seems to be of independent interest. Along the way we also point out that Gavruta’s example extends to the class of (ρ,N)-contractions. Namely, let C∞,∞(H) := [∪,NC,N(H); then, for any integer p > 1, there exists an operator T such that Tp = I and T ∉ C∞,∞(H).