On the Structure of Coisometric Extensions

Abstract

If T is a bounded linear operator on a Hilbert space H and V is a given linear isometry on a Hilbert space K, we present necessary and sufficient conditions on T in order to ensure the existence of a linear isometry π:H→K such that πT=V*π (i.e., (π,V*) extends T). We parametrize the set of all solutions π of this equation. We show, for example, that for a given unitary operator U on a Hilbert space E and for the multiplication operator by the independent variable Mz on the Hardy space HD2(D), there exists an isometric operator π:H→E⊕HD2(D) such that (π,(U⊕Mz)*) extends T if and only if T is a contraction, the defect index δT≤dimD and, for some Y:AT→E, (Y,U*) extends the isometric operator AT1/2h↦AT1/2Th on the space AT=ATH¯, where AT is the asymptotic limit associated with T. We also prove that if T is isometric and V is unitary, there exists an isometric operator π:H→K such that (π,V) extends T if and only if (a) the spectral measures of the unitary part of T (in its Wold decomposition) and the restriction of V to one of its reducing subspaces K0 possess identical multiplicity functions and (b) dim(kerT*)=dim(K1⊖VK1) for a certain subspace K1 of K that contains K0 and is invariant under V. The precise form of π, in each situation, and characterizations of the minimality conditions are also included. Several examples are given for illustrative purposes.