CONVERGENCE PROPERTIES OF THE ALUTHGE SEQUENCE OF WEIGHTED SHIFTS

摘要

For any bounded linear operator T on a Hilbert space H, the Aluthge transform is defined by Delta(T) =vertical bar T vertical bar(1/2) U vertical bar T vertical bar(1/2), where T = U vertical bar T vertical bar is the polar decomposition of T. In this paper we address a conjecture by Jung, Ko, and Pearcy [4] who proposed that for every bounded operator T, the iterates of the Aluthge transform T, Delta(T),Delta(2)(T), ... converge to a quasinormal operator. We show for any weighted shift operator T-omega with a weight sequence omega that is eventually bounded away from zero, the Aluthge sequence either converges in the strong operator topology sense or diverges to an "interval" of shift operators, in that the set of all SOT subsequential limits of {Delta(n)(T-omega)}(n=1)(infinity) has the form {t . SP; a <= t <= b} for some a <= b. Here S is the pure forward shift, and P is a projection that depends on omega. So, while the sequence may fail to converge, the smoothing provided by repeatedly applying the Aluthge transform does only produce quasinormal subsequential limits.