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Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee, Quoc-Phong Vu (2014)

Studia Mathematica

We consider simultaneous solutions of operator Sylvester equations A i X - X B i = C i (1 ≤ i ≤ k), where ( A , . . . , A k ) and ( B , . . . , B k ) are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and ( C , . . . , C k ) is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of ( A , . . . , A k ) and ( B , . . . , B k ) do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

Sobre la ecuación cuadrática en operadores A + BT +TC + TDT = 0.

Vicente Hernández, Lucas Jódar (1983)

Stochastica

By means of the application of annihilating entire functions of an operator, the bilateral quadratic equation in operators A + BT +TC + TDT = 0, is changed into an unilateral linear equation, obtaining conditions under which the solutions of such linear equation satisfy the quadratic equation.

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