Mosaic and trace formulae of log-hyponormal operators. (Q1396414)

Let \(H\) be a complex Hilbert space and \(B(H)\) the algebra of all bounded linear operators on \(H\). An operator \(T\in B(H)\) is \(p\)-hyponormal if \((T^* T)^p\geq (TT^*)^p\). An operator \(T\in B(H)\) is log-hyponormal if \(T\) is invertible and \(\log(T^*T)\geq \log(TT^*)\). \textit{K. Tanahashi} gave an example of a log-hyponormal operator which is not \(p\)-hyponormal in [Integral Equations Oper. Theory 34, 364--372 (1999; Zbl 0935.47015)]. In the present paper, the authors establish the existence of a mosaic for log-hyponormal operators as the singular integral model [see \textit{D. Xia}, ``Spectral theory of hyponormal operators'' (Operator Theory: Advances and Applications 10, Birkäuser-Verlag, Basel) (1983; Zbl 0523.47012)]. Also, they prove a Helton-Howe type trace formula for log-hyponormal operators [see \textit{M. Martin} and \textit{M. Putinar}, ``Lectures on hyponormal operators'' (Operator Theory: Advances and Applications 39, Birkhäuser-Verlag, Basel) (1989; Zbl 0684.47018)].