Positive solutions of a class of matrix equations were studied by Bhatia, et al., Bull. London Math. Soc., 32, 214 (2000), SIAM J. Matrix Anal. Appl., 14, 132 (1993) and 27, 103–114 (2005), by Kwong, Linear Algebra Appl., 108, 177–197 (1988), and by Cvetković and Milovanović, [Linear Algebra Appl., 429, 2401–2414 (2008)]. Following the idea used in the last paper, we study a class of operator equations in infinite-dimensional spaces and prove that the positivity of solutions can be established for this class of equations under the condition that a certain rational function is positive semidefinite.
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References
R. Bhatia, Positive Definite Matrices, Princeton Univ. Press, Princeton; Oxford (2007).
R. Bhatia and C. Davis, “More matrix forms of the arithmetic-geometric mean inequality,” SIAM J. Matrix Anal. Appl., 14, 132–136 (1993).
R. Bhatia and D. Drisi, “Generalized Lyapunov equation and positive-definite functions,” SIAM J. Matrix Anal. Appl., 27, 103–114 (2005).
R. Bhatia and K. R. Parthasarathy, “Positive-definite functions and operator inequalities,” Bull. London Math. Soc., 32, 214–228 (2000).
M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators on the Hilbert Space, Reidel, Dordrecht, etc. (1986).
V. I. Bogachev, Measure Theory, Springer, Berlin–Heidelberg (2007), Vol. 1.
A. S. Cvetković and G. V. Milovanović, “Positive definite solutions of some matrix equations,” Linear Algebra Appl., 429, 2401–2414 (2008).
L. Dai, “Singular control systems,” Lect. Notes Control Inform. Sci., Springer, Berlin, etc. (1989), 118.
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, Amer. Math. Soc., Providence, R.I. (1974).
Z. Gajić and M. Qureshi, Lyapunov Matrix Equation in System Stability and Control, Academic Press, San Diego (1995).
S. K. Godunov, Modern Aspects of Linear Algebra, Transl. Math. Monogr., 17, Amer. Math. Soc., Providence, R.I. (1998).
F. Hiai and H. Kosaki, “Comparison of various means for operators,” J. Funct. Anal., 163, 300–323 (1999).
F. Hiai and H. Kosaki, “Means of matrices and comparison of their norms,” Indiana Univ. Math. J., 48, 899–936 (1999).
H. Hochstadt, Integral Equations, Wiley, New York (1973).
H. Kosaki, “Arithmetic-geometric mean and related inequalities for operators,” J. Funct. Anal., 156, 429–451 (1998).
H. Kosaki, “Positive-definiteness of functions with applications to operator norm inequalities,” Mem. Amer. Math. Soc., 212, No. 997 (2011), 80 p.
M. K. Kwong, “On the definiteness of the solutions of certain matrix equations,” Linear Algebra Appl., 108, 177–197 (1988).
P. Lancaster, “Explicit solutions of linear matrix equations,” SIAM Rev., 12, 544–566 (1970).
P. D. Lax, Functional Analysis, Wiley (2002).
A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, London (1992).
F. Smithies, Integral Equations, Cambridge Univ. Press (1958).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 2, pp. 245–260, February, 2015.
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Cvetković, A.S., Milovanović, G.V. & Stanić, M.P. Positive Solutions of a Class of Operator Equations. Ukr Math J 67, 283–301 (2015). https://doi.org/10.1007/s11253-015-1079-z
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DOI: https://doi.org/10.1007/s11253-015-1079-z