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Decomposable operators and automatic continuity. (English) Zbl 0588.47041
This paper introduces a class of decomposable operators for which it is possible to give a very useful algebraic description of the spectral maximal subspaces. This class, the super-decomposable operators, is a subset of the strongly decomposable operators.
After developing the basic theory, we relate this notion to the classical ones and present several wide classes of examples, among them multiplication operators.
This leads naturally to questions about multipliers; in Section 3 an investigation is made of some of the relationships between super- decomposability of a multiplier on a Banach algebra and the corresponding multiplication operator on the algebra of multipliers.
Finally, in Section 4 we present some applications to automatic continuity theory. We give necessary and sufficient conditions on a decomposable operator \(T\in {\mathcal L}(X)\) and a super-decomposable operator \(S\in {\mathcal L}(Y)\) that every linear map \(\theta\) : \(X\to Y\) for which \(\theta T=S\theta\) be automatically continuous. This generalizes work of B. E. Johnson and A. M. Sinclair [Trans. Am. Math. Soc. 146, 533-540 (1970; Zbl 0189.435)] and of P. Vrobová [Časopis. Pěst. Mat. 97, 142-150 (1972; Zbl 0239.47016); Czechoslovak Math. J. 23(98), 493-496 (1973; Zbl 0268.47007)].
Among the corollaries of this is the following: if \(\theta\) : \(L^ p({\mathbb{R}})\to L^ q({\mathbb{R}})\) \((1\leq p,q<\infty)\) commutes with some non-trivial translation operator, then \(\theta\) is continuous.

47B40 Spectral operators, decomposable operators, well-bounded operators, etc.