Leaf as a Poincar\'e convex domain associated with an endomorphism on a real inner product space

Abstract

We define a subset of the closure of the upper half plane associated with an endomorphism on a real inner product space, which is called the leaf. When the dimension of the space is at least 3, the leaf is a convex with respect to the Poincar\'e metric, and contains all eigenvalues with nonnegative imaginary part. Moreover, the leaf of a normal endomorphism is the minimum Poincar\'e convex domain containing all eigenvalues with nonnegative imaginary part. The most commonly studied convex domain containing eigenvalues is number range. Numerical range is convex with respect to the Euclidean metric on $\mathbb C$, so numerical range has less information than leaf about real eigenvalues. We provide a new visual approach to endomorphisms.