On the Structure of Coisometric Extensions

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