College of Liberal Arts & Sciences

# Mathematics Colloquium

**Abstract****: **

For recursively generated shifts, we present recent definitive answers about the possible equivalence of two outstanding problems in the theory of unilateral weighted shifts:

1) The Subnormality Problem: When is the Aluthge transform of a subnormal weighted shift still subnormal?

2) The Square Root Problem: When does a nonnegative Berger measure admit a nonnegative square root (for the convolution product)?

For atomic measures with at most 6 atoms, our main result provides a new analytic proof to recover known results on the equivalence of the two problems.

Subsequently, we obtain an example of a 7--atomic measure for which the equivalence fails. This provides a negative answer to a conjecture related to the above-mentioned equivalence.

*The Math Colloquium will be available live via Zoom at https://uiowa.zoom.us/j/99206823329 and recordings of the sessions will be available at https://uicapture.hosted.panopto.com/Panopto/Pages/Sessions/List.aspx?folderID=b23b61cc-a48a-4de7-99fc-ae46015b8b6e usually within a couple hours of the colloquium finishing.

Fyi.: We usually gather on colloquium Thursdays at 2:30 pm in the common room, Muhly lounge, so folks can meet the speaker.

And after the talk we also plan colloquium dinner, usually at 6:30 pm. Everyone is welcome to all the colloquium events.