The homogenous compression of a bivariate weighted shift to a unilateral weighted shift

Professor Raúl Curto

Abstract: For every bivariate weighted shift W_(\alpha,\beta) acting on l^2(Z_+^2), we use the homogeneous decomposition of Z_+^2 to define a naturally associated
unilateral weighted shift W_\omega. This induces a canonical map \Phi: BWS \rightarrow UWS between the classes of bivariate and unilateral weighted shifts, which preserves such properties as hyponormality, subnormality, and spherical quasinormality. In particular, if W_(\alpha,\beta) is subnormal with Berger measure \mu, then \Phi(W_(\alpha,\beta)) is also subnormal, and its Berger measure is $\mu (phi ^{-1}), where \phi :{R}_+^{2}\rightarrow \mathbb{R}_+ is given by \phi ( s,t ) :=s+t, with {R}_+:=[0,+\infty).

As an application, we present a new, straightforward proof that the Drury-Arveson bivariate weighted shift is not hyponormal. In addition, we exhibit examples showing that \Phi is a left inverse to maps from $UWS to BWS describing well-known embeddings of unilateral weighted shifts in 2-variable weighted shifts.

Our results establish a new two-way bridge between {BWS} and UWS.

The talk is based on joint work with Jasang Yoon (University of Texas Rio Grande Valley).

To participate in this event virtually via Zoom, go to https://uiowa.zoom.us/j/95316149275.