Decomposable Operators and Generalized Intertwining Linear Transformations

Given a pair of continuous linear operators T ε L(X) and S ε L(Y) on complex Banach spaces X and Y, respectively, it is natural to ask for necessary and sufficient conditions on T and S, which force every linear transformation Θ: X → Y satisfying S Θ = ΘT to be continuous. This automatic continuity problem for intertwining linear transformations dates back to Johnson and Sinclair [14], [15], [20] and has received some considerable attention thereafter; see for instance [3], [11], [13], [21], [24]. In the rather general context of decomposable operators T and S, it has been observed only recently that this problem is closely related to the algebraic representation of the spectral maximal spaces of certain decomposable operators [2], [17], [18].