Abstract. Regular subspaces are tensor products of subspaces. The structure of regular subspaces that are invariant or reducing for the tensor product of a finite collection of Hilbert space operators is entirely characterized. Necessary and sufficient conditions for a multiple tensor product of operators to be a unilateral shift are established, and it is proved that a multiple tensor product of operators is a completely nonunitary contraction if and only if each factor is a contraction, one of them being completely nonunitary.

AMS Subject Classification (1991): 47A80, 47A15

Keyword(s): tensor product, Hilbert space operators, invariant subspaces

Received October 1, 2008, and in final form April 8, 2009. (Registered under 6435/2009.)