# On some generalizations of the numerical range and their properties

## Description

For a bounded linear operator A (or, in the finite dimensional setting, an n-by-n matrix A) its classical numerical range is defined as the mage of the unit sphere under the mapping f(x)=(Ax,x). The normalized numerical range results when in this definition (Ax,x) is replaced by (Ax,x)/||Ax||. We will discuss some general properties of this set, and in particular provide its complete description in the cases when A is normal or n=2.

## Date of Event

January 17, 2019

## Location

Parker Building, Room 301

On some generalizations of the numerical range and their properties

Parker Building, Room 301

For a bounded linear operator A (or, in the finite dimensional setting, an n-by-n matrix A) its classical numerical range is defined as the mage of the unit sphere under the mapping f(x)=(Ax,x). The normalized numerical range results when in this definition (Ax,x) is replaced by (Ax,x)/||Ax||. We will discuss some general properties of this set, and in particular provide its complete description in the cases when A is normal or n=2.

## Presenter Bio

Since 1990 and before moving to NYUAD, Ilya Spitkovsky was Professor of Mathematics at the College of William and Mary (Williamsburg, Virginia), where he has taught a wide range of courses, including linear algebra; real, complex and functional analysis. His research is mostly in Analysis, in particular, Operator and Matrix Theory. He has authored more than 200 publications, including four research monographs, and was the editor of four more. Ilya Spitkovsky is currently a member of the editorial board of five international journals.