About the characterization of the operator spectrums

Hello,

I've a question about the operator spectrum.
It is well known that the spectrum of a bounded operator on a Banach space is a non empty compact set of the complex field, so, given an operator T there exists a compact set S(T). Is it true the converse? I.e., given a compact set K, can we find a Banach space X and a bounded operator T such that T has spectrum K? If the answer is NO, can it be characterized the compact sets K such that they are spectrum sets?

Thank you to all,

Jordi-Lluís Figueras Romero
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