Example of a compact set that isn't the spectrum of an operator This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity:

Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty compact set $K \subset \mathbb{C}$ such that $K$ is not the spectrum of a bounded linear operator $T: X \to X$?

As I'm not at all knowledgeable beyond the the very first basics in the geometry of Banach spaces I apologize if the following notes completely miss the point. I add them to give some background and in order to clarify what I consider "easy":
Notes:


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*If $X$ is a Hilbert space (or more generally, if $X$ admits an unconditional basis $\{e_{n}\}$), it is easy to construct a diagonal operator with spectrum $K$ by choosing a countable dense subset $\{\lambda_{n}\} \subset K$ and letting $T$ be the diagonal operator sending $\sum x_n e_n$ to $\sum (\lambda_n x_n)e_n$.

*Standard examples for spaces without an unconditional basis are $L^1[0,1]$ and $C[0,1]$. I think I convinced myself that in both cases every non-empty compact set of $\mathbb{C}$ arises as the spectrum of an operator, so these obvious candidates don't seem to answer my question. (If this should be wrong, please tell me!)

*Variants of the Gowers-Maurey space and the Argyros-Haydon space afford examples such that the spectrum $K$ must be countable with at most one accumulation point. See Gowers's blog for background on that. For the Argyros-Haydon space this is easy to see by the very motivation for its construction: It has the remarkable property that a bounded linear operator is of the form $\lambda \cdot \operatorname{id} + C$, where $C$ is compact (thus solving the long-standing scalar-plus-compact problem).

*I asked a version of this question a few weeks ago on math.SE but with no answers so far. In view of the illuminating comments by Robert Israel and Jonas Meyer I got there I updated it a bit.

*The present question is related to Pietro Majer's question Banach spaces with few linear operators ? here on MO. I looked at Maurey's chapter Banach spaces with few operators in the Handbook of the Geometry of Banach spaces Vol. 2, Elsevier 2003, (Johnson, Lindenstrauss, eds) but the examples discussed there are way beyond what I would count as  easy.

*It may well be (as I'm rather ignorant on this topic) that the level of difficulty of an example must be comparable to the one of the construction of the Gowers-Maurey space or even the Argyros-Haydon space, so if there's a compelling reason pointing in this direction, please let me know.
 A: I don't know if this question is still interesting for someone, but in 2008 Argyron and Haydon constructed in "A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem. Acta Math. 206 (2011), no. 1, 1–54" a infinite-dimensional Banach space in which every operator is the sum of a compact operator and a multiple of the identity. Since the spectrum of compact operators consist of a sequence of point that accumulates only at zero and a escalar perturbation only translate this spectrum, I think this kind of space is pretty suitable for your cuestion. Just take a connected compact set $K$ that is not a singleton and you cannot have operators $T$ with $\sigma(T) = K$. Moreover, it is enough your compact set to have two or more accumulation points to have this property. 
It is interesting to mention that one of the other "applications" these space has is to solve a conjectute about the invariant subspace problem. By Lomonosov theorem every compact operator has a non-trivial hyperinvariant subspace, so every escalar perturbation of a compact operator also has a hyperinvariant subspace too. This is the first non-trivial example of an infinite-dimensional space such that every operator has a non-trivial hyperinvariant susbpace.
