On the characterization of 2 x 2 rho-contraction matrices

We give an explicit description of all rho -contractive (in Nagy-Foias sens e) 2 x 2 matrices. This description leads to the formulas for rho -numerica l radii when the eigenvalues of such matrices either have equal absolute va lues or equal (mod pi) arguments. We also discuss (natural) generalizations to the case of decomposable operators with at most two-dimensional blocks covering, in particular, the quadratic operators. (C) 2001 Elsevier Science Inc. All rights reserved. AMS classification: Primary 47A12; Secondary 15A 60; 47A420.