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Fractional powers of hyponormal operators of Putnam type. (English) Zbl 1096.47026

Compared to the theory of bounded hyponormal operators, very little is known about unbounded ones. The present note delivers a welcome contribution in this direction. Let \(K\) be a self-adjoint, possibly unbounded operator and let \(L\) be a bounded self-adjoint operator. Then \(A=K+iL\) is hyponormal if
\[ \| A^\ast x \| \leq \| Ax \| , \;x \in \text{Dom}(A). \]
Note that \(\text{Dom}(A) = \text{Dom}(A^\ast)\) in this case. The author considers two such hyponormal operators \(A\) and \(B\) and the delicate question of estimating the domains of the closures of \((A+B)^\alpha\) and \((A+B)^{\ast \alpha}\), where \(0<\alpha<1\). This is done under some sectoriality assumptions. The main result can be considered as a variation on Kato’s square root problem. An application is included at the end of the note: a non-symmetric, relatively bounded, sectorial perturbation \(T\) of the Schrödinger operator on the whole space \(\mathbb R^d\). An accurate computation of the domains of the fractional powers \(T^\alpha\) is sketched.

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47B25 Linear symmetric and selfadjoint operators (unbounded)
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