Projections with respect to various norms

10/14/2011 - 1:30pm
10/14/2011 - 2:30pm
Asuma Aksoy (Claremont McKenna College)

Let $ B(X) $ be a Banach algebra of all continuous linear operators on a Banach space $ X $ and $ T\in B(X) $.
 For $ (x,y)\in B(X) \times B(X^*) $, the operator norm of $ T $ is defined as

$$\|T\|=\sup|\langle Tx,y \rangle|,$$

while its numerical radius

$$\nu(T)=\sup |\langle Tx,y \rangle|$$

under the added condition that $ \langle x,y \rangle = 1 $.

The interplay between  $ \|T\| $ and $ \nu(T) $ has been the subject of much research since Bauer's definition of numerical range in the $ 1960 $'s. After pointing out some major results in this area, I discuss how extensions of operators, in particular the minimality of projections, can be measured with respect to numerical radii.

Millikan 213, Pomona College
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