Polynomially unbounded product of two polynomially bounded operators

Abstract

where S is a unilateral forward shift of infinite multiplicity, S∗ is its adjoint, and Yα is a matrix (αi+jCi+j ), with {αk}k∈N a sequence of complex numbers, and {Ck}k∈N a family of operators satisfying the “canonical anticommutation relations”. We exhibit a sequence {αk} such that T is polynomially bounded but T⊗T is not. This shows that the product of two commuting polynomially bounded operators need not be polynomially bounded itself.