The solutions to some operator equations in Hilbert $C^*$-module

Subject Areas : Functional analysis

M.  Mohammadzadeh Karizaki ¹ (Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran)
M.  Hassani ² (Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran)

Received: 2015-11-28 Accepted : 2015-03-27 Published : 2015-04-01

Keywords: Operator equation, Moore-Penrose inverse, Complemented submodule, Closed range, Hilbert C*-module,

Abstract :

In this paper, we state some results on product of operators with closed rangesand we solve the operator equation $TXS^*-SX^*T^*= A$ in the general setting of theadjointable operators between Hilbert $C^*$-modules, when $TS = 1$. Furthermore, by usingsome block operator matrix techniques, we find explicit solution of the operator equation$TXS^*-SX^*T^*= A$.

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