Research On the Positive Operator Solutions to an Operator Equation X − A ∗ X − t A �

cited. In this paper, the positive operator solutions to operator equation X − A ∗ X − t A � Q ( t > 1) are studied in inﬁnite dimensional Hilbert space. Firstly, the range of norm and the spectral radius of the solution to the equation are given. Secondly, by constructing eﬀective iterative sequence, it gives some conditions for the existence of positive operator solutions to operator equation X − A ∗ X − t A � Q ( t > 1). The relations of these operators in the operator


Introduction
Let B(H) be the set of all bounded linear operators on the Hilbert space H.In this paper, we consider the nonlinear operator equation where ,  ∈ B(H),  > 0, and  is an unknown operator in B(H).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][7][8][9][10][11][12][13]), and different iterative methods for computing the positive definite solutions to (1) are proposed in finitedimensional space.The case  =  = 1 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  ∈ B(H), if (, ) ≥ 0 for all  ∈ H, then  is said to be a positive operator and is denoted by  ≥ 0. If  is a positive operator and invertible, then denote  > 0.
For  and  in B(H),  ≥  means that  −  is a positive operator.For  ∈ B(H),  * , (), (), and () denote the adjoint, the radius of numerical range, the spectrum, and the spectral radius of , respectively.
For positive operators in B(H), the following facts are well known.
(3) If the sequence {  } +∞ =1 of positive operator is monotonically increasing and has upper bound, that is,   ≤  +1 ≤  1 , or is monotonically decreasing and has lower bound, that is,  2 ≤  +1 ≤   , then this sequence is convergent to a positive limit operator, where  1 ,  2 ∈ B(H) are given operators.

Main Results and Proofs
In order to prove our main result, we begin with some lemmas as follows.
In this section, we give our main results and proofs.

Theorem 6 .
(12)he operator equation (1) has a positive operator solution , then it has a maximal one   .Moreover, the sequence {  } in(12)for  ∈ [α, 1] is monotonically decreasing and converges to   , where α is defined in Theorem 5.