Semigroups, functional models and Hankel operators
by
Jonathan R. Partington
University of Leeds

Let T(.) be a C₀ semigroup on a Hilbert space, and let A denote its infinitesimal generator, with domain D(A). A Hilbert space operator C: D(A) --> Y is said to be admissible for T(.), if there is a constant m > 0 such that || CT(.)x ||_{L²(0, \infty; Y)} <= m ||x|| for all x in D(A). A much-studied conjecture of George Weiss asserts that C is admissible if and only if there is an M > 0 such that ||C(sI-A)^-1|| <= M/ \surd{Re s} for all s in the open right half-plane.

The conjecture is known to hold for contraction semigroups in the case that Y is finite-dimensional (and can be proven using the Sz.-Nagy-Foias functional model); this result contains the Carleson embedding theorem and the Bonsall theorem on the boundedness of Hankel operators as special cases, in an elementary way. However, the conjecture fails for the right shift semigroup on L²(0, \infty) if Y is infinite-dimensional, a result which may be expressed in terms of conditions for boundedness of vector-valued Hankel operators.

We review the above results, and certain more recent developments concerning admissibility and Hankel operators. This is joint work with Birgit Jacob and Sandra Pott.

Date received: May 29, 2002