The Numerical Range Operators In Finite Dimensional Hilbert Space

Operator numerical ranges and operator matrix are the most active research topics in operator theory in recent years.Toeplitz and Beuer first introduced the numerical range of Hilbert space and Banach space respectively in 1918 and 1962.Since the convexity theorem of the numerical range was firstly proved by Toeplitz and Hausdorff,research on numerical ranges continues to be problematic and gradually becomes multiple branches of mathematics.This article mainly investigated the calculating of operator's numerical range in Hilbert space of three dimension.Operator block techniques was introduced in this research.The content of this article involved two-dimensional Hilbert space,three-dimensional Hilbert space and other finite-dimensional Hilbert space.Furthermore,this article consisted of the basic properties and the basic theorem of the numerical range of operators.The numerical ranges of some special matrices and their properties were also studied,such as numerical ranges and affine transformation of idempotent operators,Hermitian matrices and other general matrices.