Mosaic and principal functions of log-hyponormal operators (Q1880347)

\textit{D. Xia} [``Spectral theory of hyponormal operators'' (Oper. Theory, Adv. Appl. 10, Birkhäuser Verlag, Basel) (1983; Zbl 0523.47012)] studied the mosaic and the principal function of semi-hyponormal operators with equal defect and nullity. A bounded linear operator \(T\) defined on a complex Hilbert space \(\mathcal{H}\) is said to be log-hyponormal if \(T\) is invertible and satisfies \(\log (T^*T)\geq \log (TT^*)\). In [Integral Equations Oper. Theory 34, No. 3, 364--372 (1999; Zbl 0935.47015)], \textit{K.~Tanahashi} showed that if \(T=U| T| \) is the polar decomposition of a bounded operator \(T\), then the Aluthge transform \(T_1=| T| ^{\frac12}U| T| ^{\frac12}\) of \(T\) is a semi-hyponormal operator. Using this, the authors of the paper under review define the mosaic and the principal function of a log-hyponormal operator as the mosaic and the principal function of its Aluthge transform. The aim of the present article is to study some properties of mosaic and the principal functions of log-hyponormal operators and the determining sets connected with log-hyponormal operators.