A parametrization for the symbols of a Hankel type operator

Abstract

Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered. In this more general framework, a linear operator X from a Hilbert space H1 to a Hilbert space H2 is said to be a Hankel operator for given contractions T1 on H1 and T2 on H2 if, and only if, XT ∗ 1 = T2X and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations V1 of T1 and V2 of T2. A Hankel symbol of X is a dilation Z of X, with a certain norm constraint, such that ZV ∗ 1 = V2Z. The boundedness condition imposed to X has revealed to be essential, indeed necessary and sufficient, for X to admit Hankel symbols. As for a description of the symbols of X, this work provides a parametric labeling of all of them by means of Schur like formula. As a by-product, a new proof of the existence of Hankel symbols is obtained. The proof is established by associating to X, T1 and T2 a suitable isometry V so that there is a bijective correspondence between the symbols of X and the family of all minimal unitary extensions of V.