The K-Numerical Range Of Operator And A Pair Of Orthogonal Projections

In this article, we study the k- numerical range of operator, the k- numerical range of compact operator and a pair of orthogonal projections, to which much attention is paid by many scholars in the field of operator theory. We divide the article into four chapters to study them.Firstly, we give some basic knowledge in order to discuss questions conveniently which are mentioned above. Then, beginning with the famous theorem in the field of numerical range - Hausdorff-Toeplitz Theorem (mentioned in article [16]), we study the primitive properties of the k- numerical range and get some special characters of the k- numerical range in the second chapter of this article. Moreover, combining with some parts of the special properties of extreme points, we discuss and characterize the extreme point of the k- numerical range in the second part of this chapter.It is well-known that linear compact operators on a Hilbert space form one of the most important classes of bounded operators and have many excellent properties. Based on these properties, we study k- numerical range of compact operators in detail and give the k- numerical range's prescription of compact operator and trace-class operator in the first part of the third chapter of this article, respectively. Moreover, the following results are proved: (1) Let T ∈ B(H). Then T is a compact operator if and only if .(2) Let T ∈ B(H). Then T is a trace-class operator if and only if Wk(T) is bounded. On the other hand, M. T. Chien, Shu-Hsien and Pei Yuan Wu have given the geometric properties of k- numerical ranges of quadratic operators (satisfying T2 + aT + bl = 0 for some scalars o and b in C). In the third chapter of this article, beginning with the Hausdorff-Toeplitz Theorem, we research the k- numerical range of compact operator and partial content of the k- numerical range of quadratic operator, and get some important and useful properties of them.The orthogonal projection is one of the most important classes of bounded operators, too. And orthogonal projection arise in a variety of applications, such as numerical analysis (e.g the method of approximation) and matrix theory. In recent years, J. Avron, R. Drnoverk, J. Grob, J. Baksalary and other many scholars have studied the product of a pair of orthogonal projections and the commutator of a pair of orthogonal projections in many articles (such as [2], [4], [8] and [14]), which have just been restricted on the fini te-dimensional Hilbert space. In the fourth chapter of this article, we study the product and the commutator of a pair of orthogonal projections on the a separable complex Hilbert space. Moreover, we characterize the product of a pair of orthogonal projections and prove the following result:(1) Let P1,P2 ∈ P(H] and P(m,l) denote an m-factor product of P1 and P2 with PI being the first factor and P1, P2 occurring alternately (l,j = 1,2;l @ j). Then the following statements are equivalent.(i) P(m,l) = P(n,j) some m, n > 2 and l,j = 1,2 (except for the trivial case where simultaneously m = n and j = /);(ii) P1P2 = P2F1;(iii) P(miJ) = P(nJ) for any m,n > 2 and l, j = 1,2.