Aluthge transforms of essentially normal operators
Dongjun Chen · Journal of Shandong University · 2009
Let T∈B(H), T=U|T| be a polar decomposition of T, then Tt～=|T|tU|T|1-t and Tt～(*)=|T*|tU|T*|1-t were defined to be the generalized Aluthge transform and the generalized *-Aluthge transform of T respectively, t∈(0,1). Using block operator matrices and operator approximation, the relationships among the generalized Aluthge transform, the genralized *-Aluthge transform and the class of essentially Normal Operators were discussed. In this note, it was proved that both the generalized Aluthge transform and the generalized *-Aluthge transform preserve essential normality. Moreover, it was shown that T, Tt～and Tt～(*) are mutually essentially unitary equivalent.