Aluthge transforms of weighted composition operators in $L^2$-spaces

The talk is aimed at presenting recent results concerning Aluthge transforms of (unbounded) weighted composition operators acting in $L^2$-spaces. Recall that, given a $\sigma$-finite measure space $(X,{\mathscr A}, \mu)$, an ${\mathscr A}$-measurable transformation $\phi$ of $X$ and a complex ${\mathscr A}$-measurable function $w$ on $X$, the weighted composition operator in $L^2(\mu)$ induced by $\phi$ and $w$ is given by $$ \mathcal { D}({C_{\phi,w}} ) = \{f \in L^2(\mu) \colon w \cdot (f\circ \phi) \in L^2(\mu)\},$$ \[ {C_{\phi,w}} f = w \cdot (f\circ \phi), \quad f \in \mathcal { D}(C_{\phi,w}), \] We will show that the $\alpha$-Aluthge transform $\Delta_\alpha({C_{\phi,w}})$ of a densely defined weighted composition operators ${C_{\phi,w}}$ is a closable operator whose closure is a weighted composition operator $C_{\phi,w_\alpha}$ induced by $\phi$ and a weight function $w_\alpha$ that can be written in terms of the transformation $\phi$, the weight function $w$, and the Radon-Nikodym derivative ${{\mathsf h}_{\phi,w}}$, canonically attached to ${C_{\phi,w}}$. We will supply conditions for the equality $\Delta_\alpha({C_{\phi,w}})=C_{\phi,w_\alpha}$. We will provide a characteriaztion for $p$-hyponormality of unbounded weighted composition operators in $L^2$-spaces and presents results concerning $p$-hyponormality of Aluthge transforms of weighted composition operators. The talk is based on a joint work with C. Benhida, J. Stochel, and J. Trepkowski.