\(\lambda\)-Aluthge transforms and Schatten ideals
Let \(T\) be a bounded linear operator on a Hilbert space and let \(T=U| T| \) be the polar decomposition of \(T\). Then for every \(\lambda \in [0,1]\) the \(\lambda\)-Aluthge transform is defined by \(\Delta_\lambda(T)=| T| ^\lambda U | T| ^{1-\lambda}\). For \(\lambda=1/2\), this gives the usual Aluthge transform introduced in [\textit{A. Aluthge}, Integral Equations Oper. Theory 13, No. 3, 307--315 (1990; Zbl 0718.47015)] to study \(p\)-hyponormal operators. The authors prove generalizations of well-known properties of the Aluthge transform to \(\lambda\)-Aluthge transforms, as well as several results which are new even for the usual Aluthge transform. In particular, they show that for \(\lambda \in (0,1)\), the map \(T \mapsto \Delta_\lambda(T)\) is continuous at every closed range operator \(T\). Moreover, for \(\lambda \in (0,1)\), they prove that the Schatten \(p\)-norms of the \(\lambda\)-Aluthge transforms decrease with respect to the Schatten \(p\)-norms of the original operator, and that equality holds if and only if \(T\) is normal. The relationship to Riesz's functional calculus is also considered as well as the Jordan structure of the iterated \(\lambda\)-Aluthge transforms in the finite-dimensional case.
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