Composition operators and weighted shifts

01/15/2018 - 4:15pm
01/15/2018 - 5:15pm
George Exner (Bucknell)

Two standard presentations of Hilbert space are as a space of square summable complex-valued sequences and as square integrable analytic functions on the open unit disk $\mathbb{D}$.  A standard example of a bounded linear operator on the first is a weighted shift, and on the second the composition operator $C_\varphi$ associated with an analytic function $\varphi$ mapping $\mathbb{D}$ to $\mathbb{D}$.  In work joint with Isabelle Chalendar, we exhibit some composition operators with invariant subspaces on which they act as weighted shifts.  Consideration of the subnormality of the shift (an ``operator theory question'') gives information about a fixed point of $\varphi$ (a ``function theory question''), and the results yield also some intriguing complex geometry which we do not understand well.

Emmy Noether Rm Millikan 1021 Pomona College

Claremont Graduate University | Claremont McKenna | Harvey Mudd | Pitzer | Pomona | Scripps
Proudly Serving Math Community at the Claremont Colleges Since 2007
Copyright © 2018 Claremont Center for the Mathematical Sciences