A Characterization Of Non-extreme Points Of Numerical Ranges And Researches On Relative Problems

In 1918 O. Toeplitz gave a new concepts on complex matrix: numerical range and in 1919 F. Hausdorff proved that the numerical range of a complex matrix is convex. After that, the researches on the geometric properties of numerical range ([21][30][22][6][21]) become active. The subject is related and has applications to many different branches of pure and applied mathematics such as functional analysis, system theory, quantum physics and so on.As a result of the development of numerical range, various genelized numerical range, such as joint numerical range, c-numeical range, maximal numerical range, essential numerical range and essential maximal numerical range have been studied. The quadratic numerical range, one of the important generalizations of the numerical range, was put forward by Lange and Treter in the course of studying of the spectral theory of the block operator matrix. And in 2001, Lange, Markus, Matsaev and Treter gave the elementary properties of quadratic numerical range, and also they discussed the coner points of numerical range and quadratic numerical range.There is a bijection between the set of all the projections on Hilbert space H and the set of all the closed subspaces of Hilbert space H. [19] showed that R(A) + R(B) = R((AA* + BB*} 1/2). Generic pair of two subspaces was studied in [24]. [12] gave a characterization of generic pair of two subspaces. In this paper, non-extreme points of numerical range, n-numerical range and orthogonal projection pair are discussed, the main content as follows:First, in Chapter One we give the signs, definitions and elementary theorems used by this article.In the second section of Chapter Two, we give a characterization of non-extreme points of numerical range. If we know the extreme points of numerical range, then we can characterize numerical range. Conversely, if we know the non-extreme points of numerical range, then we can also characterize numerical range. We just discuss the properties of non-extreme points of numerical range in this article.And in Section Three, we study a generized numerical range: n-numerical range. It is well known that it can locate the spectrum better than numerical range. We discusscuss the relations between spectrum and n-numerical range of compact operators and getIn the third chapter of this article, we discuss the properties of sum, difference and product of an orthogonal projection pair (P, Q). We also give a characterization that the orthogonal projection pair (P, Q) is Predholm and discuss the properties of the spectrum and norm of the commuter A := PQ - QP.