Some properties for the existence of a positive definite solution of matrix equation \(X+A^*X^{-2^m}A=I\). (English) Zbl 1031.15015

Let \(P(n)\) be the set of all positive definite \((n\times n)\) matrices; \(I\) is the \((n\times n)\) identity matrix; and \(m, n\) \(\in \mathbb{N}\). The authors studies the matrix equation \[ X+A^*X^{-2^m}A=I \tag{1} \] with given matrix \(A\in P(n)\) and unknown matrix \(X\in P(n)\). They give necessary and sufficient conditions for the solvability of equation (1). The main results of this paper are:
Theorem 3. If there exist numbers \(a, b\) satisfying \(0<a<b<1\) and the inequalities \[ a^{2m}(1-a)I<AA^*<b^{2m}(1-b)I \] hold, then (1) has a positive definite solution.
Theorem 6. If (1) has a positive definite solution \(X\), then \[ A^*A+ (AA^*)^{\frac{1}{2m}}<I. \]


15A24 Matrix equations and identities
15B48 Positive matrices and their generalizations; cones of matrices
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