Intertvining. Operators

In this thesis we study operators, and spaces of operators on a Hubert space defined by intertwining relations. The classical Hankel operators are those operators which intertwine the unilateral shift and its adjoint. We consider generalised Hankel operators relative to shifts and relative to families of shifts and give generalisations of the classical theorems of Nehari and Hartman. In contrast to the classical approach our proofs are mainly geometric and rest on the Sz-Nagy Foias lifting theorem. We show that the closed linear span of the positive Hankel operators is a proper subspace of the Hankel operators and contains all the compact Hankels. Part of this result is also obtained.,via Douglas's localisation theory for Toeplita operatorsfrom the fact that there exist, Harikel operators which do not lie in the C*_a].gebra generated by the Toeplitz operators. In chapter 7 we see that certain sums of spaces of intertwining operators are closed and yield C*_algebras. In fact it is the algebraic properties of these spaces that ensure the automatic closure of their sum. As a consequence we obtain odd/even decompositions for C-algebras and von Neumann algebras and related double. commutant theorems.