4 problems

Let $H$ be a separable, complex Hilbert space and let $T$ be a bounded linear operator on $H$. The iterated Aluthge transforms of $T$ are the sequence…

Let $A\in\mathbb{M}_n(\mathbb{C})$, and let $\Delta(A)=P^{1/2}UP^{1/2}$ be its Aluthge transform, where $A=UP$ is a polar decomposition with $P=|A|$. Write $\Delta^m$ for the $m$-f…

Let $A\in\mathbb{M}_n(\mathbb{C})$, and write its polar decomposition as $A=UP$, where $P=|A|=(A^*A)^{1/2}\geq 0$. For $0<\lambda<1$, define the $\lambda$-Aluthge transform by … Th…

Let $W_\mu$ be a subnormal weighted shift with finitely atomic Berger measure $\mu$. The measure $\mu$ has a square root if there exists a nonnegative measure $\nu$ such that … The…