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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!

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Not sure if you'll read this so long after first posting your question Pietro, but I just thought I'd mention that even the existence of a basis does not guarantee many operators (the Argyros-Haydon space mentioned in Yemon's answer has a basis), but certain kinds of bases do guarantee a large (i.e. nonseparable) space of operators. A somewhat recent paper on this topic is the following one by Androulakis, Beanland, Dilworth and Sanacory: Embedding $\ell_{\infty}$ into the space of bounded operators on certain Banach spaces, Bulletin of the London Mathematical Society 38 (2006) 979--990. – Philip Brooker Nov 22 '10 at 2:00
Thank you, that's a very interesting remark. – Pietro Majer Nov 22 '10 at 7:21
up vote 12 down vote accepted

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 $\ell_1$.

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The stronger "scalar + nuclear" problem seems to be unsolved... – Ady Jul 5 '10 at 21:09
With regards to Ady's comment, a space $X$ with the property that every operator is a nuclear perturbation of a scalar operator must also have the following remarkable (in tandem) properties: (i) $X$ lacks the approximation property; (ii) the finite rank operators on $X$ is a dense subset (with respect to the operator norm) of the compact operators on $X$. – Philip Brooker Jul 5 '10 at 21:23
The Argyros-Haydon construction has recently been extended by Argyros and a number of coauthors to show that if $X$ is a Banach space with norm-separable dual, and $\ell_1$ if does not embed isomorphically into $X^*$, then one may isomorphically embed $X$ in a Banach space $Z$ with $Z^*$ isomorphic to $\ell_1$ and every operator on $Z$ a compact+scalar. – Philip Brooker Jul 5 '10 at 21:34

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