An Assessment of Aluthge and Duggal Transformations: a Brief Survey |
This paper will be appeared in other journal. Letbe a separable, infinite dimensional,complex Hilbert space, and letdenote the algebra of all bounded linearoperators on. An arbitrary operator T inhas a unique polar decomposition T = UP, whereandU is a partial isometry with initial space the closure of the range of|T| and final space the closure of the range of T. Associated with Tthere is a related operator, sometimes called the Aluthge transform of T because it was studied in thecontext that T is ap-hyponormal operator (to be defined below). In this note we derive somespectral connections between an arbitraryand its associated Aluthge transform that enable us, in particular, to generalizean invariant-subspace-theorem of Berger to that context. We will also show thatthe hyperinvariant subspace problems for hyponormal and p-hyponormal operatorsare equivalent. The following lemma is completely elementary, but sets forthbasic relations between T andthat will be useful throughout the paper.