K-generalized Projections And Operator Equations

The study of operator theory began in 20th century. Since it is used widely in mathematics and other subjects, it got rapid development at the beginning of the 20th century, k-generalized projections and operator equations have become hot topics in operator theory. Let B(H) be the set of all bounded linear operators on a Hilbert space H. T ∈ B(H) is said to be a fc-generalized projection if T^k = T^*, where T^* is the adjoint of T, k ∈ N and k ≥ 2. It is well known that fc-generalized projections contain lots of elementary interesting properties of operators. In recent years, many scholars have paid much attention to the fc-generalized projections. In this article we studied the spectrum and the path connectivity of fc-generalized projections on an infinite dimensional Hilbert space. An operator equation is an equation whose entries are bounded linear operators on the corresponding Hilbert space and has played an important role on the subject of functional analysis. Since 1980s', the solution of operator equations X J - JX* = M, AX A* = B, AXB = C and AX = B on a finite dimensional space have been considered by many scholars. In this paper we continue to study these operator equations on an infinite dimensional Hilbert space, and mainly discuss some properties of the solutions to these equations. Furthermore, we study a class of special operator, i.e., generalized quadratic operator. Using operator equations, the spectrum and the group inverse of generalized quadratic operators have been studied.This paper contains four chapters.In chapter 1, we mainly introduce some notations, definitions and theorems, such as the definitions of normal operator, spectrum, invariant subspace and reducing subspace of operators, partial isometry etc. Subsequently, we give some well-known theorems such as the range inclusion theorem, polar decomposition theorem and spectral mapping theorem.In chapter 2, we discuss the fc-generalized projections on an infinite dimensional Hilbert space. In the first place, we obtain that A ∈ B(H) is a fc-generalized projection if and only if A is normal and σ (A) {0, e^{i2n/(k+1)π} n = 0,1, 2,..., k}. Secondly, we give the characterizations of fc-generalized projection A. Finally, we investigate the path connectivity of fc-generalized projections, the results as follows:(1) If P, Q e B(H) are homotopic ^-generalized projections, then P and Q are path connected;(2) There is no segment in the set of ^-generalized projections.In chapter 3, we study some properties of the solutions to operator equations X J - JX* = M, AX A* = B, AXB = C and AX = B on an infinite dimensional Hilbert space. Firstly, we give the equivalant statements that operator equation X J — JX* = M has an isometric solution;Secondly, we obtain the necessary and sufficient conditions of the solutions to the operator equation AX A* = B,AXB = C and AX — B. Moreover, the representation of the general solution to these equations is given.In chapter 4, we characterize the set of generalized quadratic operators with respect to an idemportant PC(P) = {Ae B{H) : A2 = aA + (3P and AP = PA = A,Pe P,V a,j3 e C}.We prove that the generalized quadratic operators are similarly invariant. Using the technique of operator theory, the spectrum and the group inverse of C(P) have been studied, which extends the conclusions of R. W. Farebrother and G. Trenkler in [13].