Partial isometries between Hilbert modules and their compositions

Abstract

Motivated by questions raised in the preprint1 by Accardi and Lu (private communication), we examine criteria for when the product of two partial isometries between Hilbert spaces is again a partial isometry and we use this to define a new composition operation that always yields again a partial isometry. Then, we aim at promoting these results to (not necessarily adjointable) partial isometries between Hilbert modules as proposed by Shalit and Skeide.12 The case of Hilbert spaces is elementary and rather simple, though not trivial, but – we expect – folkloric. The case of Hilbert modules suffers substantially from the fact that bounded right linear maps need not possess necessarily an adjoint. In fact, we show that the new composition law for partial isometries between Hilbert spaces can in no way be promoted directly to partial isometries between Hilbert modules, but that we have to pass to the more flexible class of partially defined isometries.