ON THE ITERATED MEAN TRANSFORMS OF OPERATORS

Abstract

Let T = U vertical bar T vertical bar be the polar decomposition of an operator T is an element of L(H). For given s,t >= 0, we say that (T) over cap (s,t) := sU vertical bar T vertical bar + t vertical bar T vertical bar U is the weighted mean transform of T. In this paper, we study properties of the k-th iterated weighted mean transform (T) over cap ((k))(s,t) of T = U vertical bar T vertical bar when U is unitary. In particular, we give the polar decomposition of such (T) over cap ((k))(s,t) and investigate its applications. Finally, we consider the iterated weighted mean transforms of a weighted shift.