What is the relation between general operator spectrum (no resriction on space nor on domain) and possibility for this operator to have square root?

17 December 2019 0 6K Report

It is known that for special cases, mainly for analytic position operators, the splitting of spectrum on two connected components ensures existence of square root (result of Gunter and Shaffer). Is This topological characterization of the spectum sufficient for general operators? If not, as predicted, what will be the additionnal hypothesis on the ambiant space and on the operator to have such property?

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