Compact and finite rank operators satisfying a Hankel type equation T2X = XT1*

In 1997, V. Ptak defined the notion of generalized Hankel operator as follo ws: Given two contractions T-1 is an element of B(H-1) and T-2 is an elemen t of B(H-2) an operator X : H-1 --> H-2 is said to be a generalized Hankel operator if T2X = XT1* and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of T-1 and T-2. Th e purpose behind this kind of generalization is to study which properties o f classical Hankel operators depend on their characteristic intertwining re lation rather than on the theory of analytic functions. Following this spir it, pie give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Ptak's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential nor m of a Hankel operator, Hartman's characterization of compact Hankel operat ors and Kronecker's characterization of finite rank Hankel operators.