5 problems

For an operator $A\in\mathcal{B}(\mathcal{H})$ with polar decomposition $A=U|A|$, define its Aluthge transform by … Write $\Delta^n(A)$ for the $n$-fold iterate, and let the strong…

Let $\mathcal{M}$ be the finite von Neumann algebra considered in the paper, with its tracial $L^2$-norm, and let $T\in\mathcal{M}$. Write $\widetilde{T}^{(n)}$ for the $n$th Aluth…

Let $\lambda\in(0,1)$ and let $T\in\mathcal{M}_r(\mathbb{C})$ have polar decomposition $T=U|T|$. Define the $\lambda$-Aluthge transform by … and let $\Delta_\lambda^\infty(T)$ deno…

Let $\mathcal{M}_r(\mathbb{C})$ denote the space of $r\times r$ complex matrices. For $T\in\mathcal{M}_r(\mathbb{C})$, let $\Delta^\infty(T)$ denote the limit of the iterated Aluth…

Constancy conjecture. The map $R_T$ is constant for every $T\in\mathcal{S}(D)$ if and only if