# Postdoc Seminar

## Superconvexity of the Spectral Radius, and Convexty of the Spectral Bound and the Type

### 何小清 博士 (华师大PDE中心博士后)

#### 2016年10月11日(周二)下午 1:00-4:00 闵行校区行政楼12楼PDE中心1202报告厅

Abstract. We prove generalizations of a theorem of J. F. C. Kingman stating that the spectral radius of a matrix whose elements are superconvex functions of a parameter $h$ is likewise superconvex in $h$, and of a theorem of J. E. Cohen to the effect that the principal eigenvalue of a real $n\times n$ matrix with all off-diagonal elements nonnegative is a convex function of its diagonal elements. These generalizations are to linear operators in an ordered Banach space $X$ with proper, closed, generating, and normal cone. Various interesting examples are given. In particular, this result can be used to study elliptic eigenvalue problems with respect to weight functions. This talk is based on the work of Kato [Math Z. 180, 265--273(1982)].