# IMath Webinar Series: Kennett Dela Rosa, Ph.D.

You are all invited to attend the second webinar of the IMath Webinar Series for this semester on Monday, October 11, 2021, 2:00pm.

Dr. Kennett L. Dela Rosa of the Matrix Analysis and Linear Algebra Group will present his work titled “Completing an Operator Matrix and the Free Joint Numerical Radius”.

** Abstract:** Ando’s classical characterization of the unit ball in the numerical radius norm was generalized by Farenick, Kavruk, and Paulsen using the free joint numerical radius of a tuple of Hilbert space operators \((X_1, \ldots , X_m)\). In particular, the characterization leads to a positive definite completion problem. In this talk, we study various aspects of Ando’s result in this generalized setting. Among other things, this leads to the study of finding a positive definite solution \(L\) to the equation \[L=I+\displaystyle\sum_{j=1}^m\left[\left(L^{\frac{1}{2}}X_j^*LX_jL^{\frac{1}{2}}+\frac{1}{4}I\right)^{\frac{1}{2}}+\left(L^{\frac{1}{2}}X_jLX_j^*L^{\frac{1}{2}}+\frac{1}{4}I\right)^{\frac{1}{2}}\right],\] which may be viewed as a fixed point equation. Once such a fixed point is identified, the desired positive definite completion is easily obtained. Along the way we derive other related results including basic properties of the free joint numerical radius and an easy way to determine the free joint numerical radius of a tuple of generalized permutations. Finally, we present some open problems.