Sometimes, dealing with the concrete and familiar Banach spaces of everyday life in maths, I happen nevertheless to ask myself about the generality of certain constructions. But, as I try to abstract such constructions up to the level of general Banach spaces, I immediately feel the cold air of wilderness coming from the less known side of this huge category. One typical difficulty is, that it's not at all obvious how to find, or to prove the existence, of (bounded linear) operators $T:E\to E.$
Certainly, there are always a lot of finite rank operators, furnished by the Hahn-Banach theorem, and their norm-limits. However, in an anonymous infinite dimensional Banach space, I do not see how to guarantee the existence of bounded linear operators different from a compact perturbation of $\lambda I:$ hence my questions:
Are there always other operators than $\lambda I+K $ on an infinite dimensional Banach space?
(in other words, can the Calkin algebra be reduced to just $\mathbb{C}\\ $)? More genarally, starting from a operator $T$, is there always a way to produce new operators different from a compact perturbations of the norm-closed algebra generated by $T$? Is there a class of Banach spaces that are rich of operators in some suitable sense (spaces with bases, for instance, but more in general?).
Thank you!
Best Answer
Examples were constructed (about two years ago?) by Argyros and Haydon. See this blog post for some non-technical discussion. It seems worth noting, as one is almost obliged to, that the space originally constructed by A & H is an isomorphic predual of $\ell_1$.