Mosaics and principal function of hyponormal and semi-hyponormal operators, Integral Equations and Operator Theory 4(1981

We extend the theory of determining functions to the con-text of hyponormal operators without requiring that T*T-TT * be in trace class. It is shown that the determining function sat-isfies a Riemann-Hilbert barrier relation, and the spectrum of T is characterized in terms of its mosaic. Inequalities re-lating the norm of the self commutator of T and the area of its spectrum are shown to fol low both in the hyponormal and the seml-hyponormal case. §i. Introduction. Results of the type of the Putnam inequality, IIT.T_TT.II< = ~'i meas c(T), for T a hyponormal operator on Hilbert space are well known; and the theme appears with many variations, such as a corresponding result related to the polar decomposition of T