In this chapter we discuss some necessary and sufficient conditions on a pair (T, R) of continuous linear operators on Banach spaces X and Y, respectively, so that each linear operator S from X into Y intertwining with T and R (i. e. satisfying ST = RS) is continuous. We begin with two results implying the existence of discontinuous intertwining operators (Section 3). The first requires the existence of a complex number μ such that μ is an eigenvalue of R and (T − μI)X is of infinite codimension in X. The second depends on there being a non-zero linear subspace Z of Y such that (R − μI)Z = Z for all complex numbers μ, and on T not being algebraic.

In Section 4 we give necessary and sufficient conditions for each linear operator S intertwining with the pair (T, R) to be continuous when the spectrum of R is countable, and in Section 5 we consider the case when T and R are normal operators on a Hilbert space.

The existence of discontinuous intertwining operators

Throughout this section X and Y are Banach spaces, and T and R are continuous linear operators on X and Y, respectively.

Definition. A complex number M is said to be a critical eigenvalue of the pair (T, R) if (T − μI)X is of infinite codimension in X and μ is an eigenvalue of R.

Lemma, If (T, R) has a critical eigenvalue, then there is a discontinuous linear operator S from X into Y intertwining with (T, R).