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# Analysis Seminar

Friday, October 9, 2015 — 3:30 PM EDT

Javad Mashreghi, Laval University

"The Halmos Conjecture on the Numerical Range"

Let $T$ be an operator on a Hilbert space $H$ with numerical radius $w(T) \leq 1$. Halmos conjectured that $w(T^n) \leq 1$, $n \geq 1$. After
several partial results, it was finally settled by Berger using dilation
theory. The Berger--Stampfli theorem, a generalization of the conjecture,
says that if $f$ is a function in the disk algebra such that $f(0)=0$,
then $w(f(T))\leq \|f\|_\infty$. We give an elementary proof of this
result using finite Blaschke products.

Joint work with H. Klaja and T. Ransford.

MC 5417

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