Constructions with the Shift Operator

Abstract

AbstractThis chapter explores three operators created from the unilateral shift S and its adjoint S*: the sum S + S*, the direct sum S ⊕ S*, and the tensor product S ⊗ S*. We give Hilbert’s spectral representation of the selfadjoint operator S + S*. The operator S ⊕ S* on H2 ⊕ H2 is complex symmetric. We examine its invariant subspaces and spectral properties. The operator S ⊗ S* on H2 ⊗ H2 is also complex symmetric. We discuss its reducing subspaces and spectral properties.