Toeplitz type operators, determining functions, principal functions, and trace formulas (Q919250)

The authors introduce an operator valued Riemann-Hilbert problem defined by symbols associated with a weak contraction T and an operator A for which \(TAT^*=A.\) The solution of this problem leads to a connection between such objects as the principal function of the hyponormal operator \(H=(s\)-lim \(T^ nT^{*n})^{1/2}T^*\) and eigenvalue distribution of \(| \theta_+(e^{it})|.\) The representation of the methods used in the paper is given by the contents: 1) Introduction. 2) Preliminary remarks. 3) Determining functions. 4) Symbols and the Riemann-Hilbert problem. 5) Principal functions associated to weak contractions and \(A_ T=s\)-lim \(T^{*n}T^ n\). 6) Phase shift formulae for dissipative operators. 7. Determinant formulae for the determining functions. 8) J-matrix representation of hyponormal oprators. 9) Comparison of the two mosaics of a hyponormal operator. 10) Hyponormal operators with rank one self- commutator and contractions t so that \(H^*=TH\).