Putnam's inequality of doubly commuting \(n\)-tuples for log-hyponormal operators (Q2732519)

Putnam's and Berger-Shaw's inequalities are among the fundamental results in the theory of semi-normal operators. Putnam's inequality asserts that the norm of the self-commutator of a hyponormal operator is majorized, up to a universal constant, by the area of the spectrum. In particular, one deduces that a pure hyponormal operator cannot have null area spectrum.NEWLINENEWLINENEWLINEAlthough a few variants of Putnam's inequality are known in the case of several commuting operators, no definitive result of this type was proved. By adapting earlier ideas of \textit{Xia Daoxing} [``Spectral theory of hyponormal operators'', Basel (1983; Zbl 0523.47012)], the authors prove several versions of Putnam type inequalities for the Taylor spectrum of doubly commuting systems of log-hyponormal operators.