Solutions to Operator Equations on Hilbert C*-Modules II

Abstract

In this paper, we study the solvability of the operator equations A * X + X * A = C and A * XB + B * X * A = C for general adjointable operators on Hilbert C * -modules whose ranges may not be closed. Based on these results we discuss the solution to the operator equation AXB = C, and obtain some necessary and sufficient conditions for the existence of a real positive solution, of a solution X with B * (X * + X)B ≥ 0, and of a solution X with B * XB ≥ 0. Furthermore in the special case that R(B) ⊆ R(A * ) we obtain a necessary and sufficient condition for the existence of a positive solution to the equation AXB = C. The above results generalize some recent results concerning the equations for operators with closed ranges. Mathematics Subject Classification (2000). Primary 46L08; Secondary 47A05.

The common positive solution to adjointable operator equations with an application

“…The positive solution to (1.1) for Hilbert space operators was derived in terms of generalized inverses in [2,3], and the results were extended to the adjointable operators acting on Hilbert C * -modules in [4][5][6]. The positive semidefinite solution to the matrix equation or the positive solution to the operator equation…”

On majorization and range inclusion of operators on Hilbert C*-modules

“…A generalization of Douglas theorem to the Hilbert C * -module case was given in [3], which can be stated as follows:…”

Operator equations AX+YB=C and AXA⁎+BYB⁎=C in Hilbert C⁎-modules

“…Generalized inverses are useful tools for investigation of solutions of operator equations in the setting of Hilbert C * -modules but these inverses need the strong condition of closedness of ranges of considered operators. Fang et al [6,7] have studied the solvability of operator equations without the closedness condition on ranges of operators by employing a generalization of a known theorem of Douglas [4,Theorem 1] in the framework of Hilbert C * -modules. In their results, concentration is based on the idea of using more general (orthogonal) projections instead of projections such as AA † .…”

“…In their results, concentration is based on the idea of using more general (orthogonal) projections instead of projections such as AA † . They investigated the equations AX = B [6, Theorem 1.1], A * XB + B * X * A = C [7, Corollary 2.8] and AXB = C [7,Theorem 3.4].…”