Suppose X1 and X2 are
complex Banach spaces with u0,⋯ , un in ℒ(X1),v ∈ℒ(X2), and suppose ⊗ is a
uniform crossnorm. The spectra of the operators ∑j=0nuf⊗ vj on X1⊗ X2 and
R : x →∑j=0nujxvf,x ∈ℒ(X2,X1), are studied in the context of a general theory.
Explicit representations are set down for the resolvents of these and more general
operators.