Automatic continuity of intertwining linear operators on Banach spaces (Q1192088)

Let \(T\in {\mathcal L}(X)\) and \(S\in {\mathcal L}(Y)\) be two bounded linear operators on Banach spaces \(X\) and \(Y\), respectively, and let \(\theta: X\to Y\) by a linear map. Denote by \(C(S,T)\) the commutator \(C(S,T)\theta=S\theta-\theta T\). The map \(\theta\) is called a generalized intertwining transformation for the pair \((S,T)\), if \(\lim_{n\to\infty} \| C(S,T)^ n \theta\|^{1/n}=0\). In this paper the following is proved: Suppose \(T\) has spectral property \((\delta)\) (\(T\) has property \((\delta)\) if and only if \(T\) is similar to a quotient of a decomposable operator), and suppose that \(S\) is semi- admissible. Then every generalized intertwining transformation \(\theta\) for \((S,T)\) is continuous if and only if the codimension of the range \((T-\lambda)X\) is finite for each eigenvalue \(\lambda\) of \(S\). Since multipliers on a commutative semisimple Banach algebra are semi- admissible, and convolution operators on group algebras are multipliers, the above result can be applied in these cases.