Subscalarity of operator transforms

Abstract

In this paper, we provide various connections between a bounded linear operator T and some of its transforms, namely the Aluthge transform (T) over tilde (A), Duggal transform (T) over tilde (D), and mean transform (T) over cap. In particular, we show that under the condition that vertical bar T vertical bar U vertical bar T vertical bar = vertical bar T vertical bar U-2 where T = U vertical bar T vertical bar is the polar decomposition, if one of T, (T) over tilde (D), and (inverted perpendicular) over tilde (A) is subscalar of finite order, then (T) over cap is also subscalar of finite order. As an application, we find subscalar operator matrices. We also give several spectral relations. Finally, we provide an equivalent condition under which a weighted shift has a hyponormal iterated mean transform. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim