Smooth operators in the commutant of a contraction

Pascale Vitse
2003 Studia Mathematica  
For a completely non-unitary contraction T , some necessary (and, in certain cases, sufficient) conditions are found for the range of the H ∞ calculus, H ∞ (T ), and the commutant, {T } , to contain non-zero compact operators, and for the finite rank operators of {T } to be dense in the set of compact operators of {T } . A sufficient condition is given for {T } to contain non-zero operators from the Schatten-von Neumann classes S p . 1. Introduction. For a given Hilbert space contraction T , we
more » ... study how "smooth" (compact, etc.) operators in the commutant {T } = {A : AT = T A} can be. The problem arises in several applications in control theory, vector-valued Hankel operators or the theory of model operators. Here it is treated in the framework of the Sz.-Nagy-Foiaş functional model and some answers are proposed in the language of the characteristic function Θ T of the contraction T . Let H be a separable Hilbert space, and L(H) the space of bounded linear operators on H.
doi:10.4064/sm155-3-4 fatcat:wkahaizjpvcmxjyxftlnby7soa