On the Positive Operator Solutions to an Operator Equation

The necessary conditions and the sufficient condition for the existence of positive operator solutions to the operator equation are established. An iterative method for obtaining the positive operator solutions is proposed. 1. Introduction Let be the set of all bounded linear operators on the Hilbert space . In this paper, we consider the nonlinear operator equation where , , and is an unknown operator in . Both and are positive integers. This type of equation often arises from many areas such as dynamic programming [1], control theory [2, 3], stochastic filtering and statistics, and so forth [4, 5]. In the recent years, for matrices, (1) has been considered by many authors (see [6–13]), and different iterative methods for computing the positive definite solutions to (1) are proposed in finite-dimensional space. The case has been extensively studied by several authors. In this paper, we extend the study of the operator equation (1) from a finite-dimensional space to an infinite-dimensional Hilbert space. We derive some necessary conditions for the existence of positive solutions to the operator equation (1). Moreover, conditions under which the operator equation (1) has positive operator solutions are obtained. Based on Banach’s fixed-point principle, we obtain the positive operator solution to the operator equation (1). First, we introduce some notations and terminologies, which are useful later. For , if for all , then is said to be a positive operator and is denoted by . If is a positive operator and invertible, then denote . For and in , means that is a positive operator. For , , , , and denote the adjoint, the radius of numerical range, the spectrum, and the spectral radius of , respectively. For positive operators in , the following facts are well known.(1)If , then .(2)Denote . Then .(3)If the sequence of positive operator is monotonically increasing and has upper bound, that is, , or is monotonically decreasing and has lower bound, that is, , then this sequence is convergent to a positive limit operator, where are given operators. 2. Main Results and Proofs In order to prove our main result, we begin with some lemmas as follows. Lemma 1. Let . If , then . Lemma 2. Let and be self-adjoint operators in . If , then, for every , one has . In this section, we give our main results and proofs. Theorem 3. If the operator equation (1) has a positive operator solution , then and , where . Proof. (1) If the operator equation (1) has a positive operator solution , then . Hence, we obtain ; that is, . From , it follows that That is, . (2) From (1), we have