Operator Theory Seminar
“Aluthge Transforms and Spherical Isometries” by Professor Raul Curto, Department of Mathematics, The University of Iowa
Abstract: For a commuting pair T=(T_1,T_2) of operators on Hilbert space, the spherical Aluthge transform of T, denoted by A(T), is defined in terms of the positive factor in the (joint) polar decomposition of the pair. A(T) is also commutative, so it admits a spherical Aluthge transform, A(A(T)). In general, the sequence of iterates do not need to converge in the weak operator topology. However, for special classes of commuting pairs, the so-called 2-variable weighted shifts, one can study in some detail the limiting behavior of the iterates of the spherical Aluthge transform. The limit pair, if it exists, is necessarily a fixed point of A, that is, a spherically quasinormal pair. For a rather large suitable class of 2-variable weighted shifts we will establish the convergence of the sequence of iterates in the weak operator topology.