Estimates for norms of matrix-valued and operator-valued functions and some of their applications. (English) Zbl 0805.47009

Summary: A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel’fand-Shilov estimate for regular functions of matrices and Carleman’s estimates for resolvents of matrices and compact operators.
From the estimates for resolvents, the well-known results for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown’s inequalities for eigenvalues of matrices are improved.
From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.


47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A55 Perturbation theory of linear operators
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A54 Matrices over function rings in one or more variables
35A30 Geometric theory, characteristics, transformations in context of PDEs
34A30 Linear ordinary differential equations and systems
Full Text: DOI


[1] Ahiezer, N. I. and Glazman, I. M.:Theory of Linear Operators in Hilbert Space, vol. 1, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1981.
[2] Bhatia, R.:Perturbation Bounds for Matrix Eigenvalues, Pitman Res. Notes Math. 162, Longman Scientific and Technical, Harlow, U.K., 1987. · Zbl 0696.15013
[3] de Branges, L.: Some Hilbert spaces of analytic functions,J. Math. Anal. Appl. 12 (1965), 149-186. · Zbl 0134.12002 · doi:10.1016/0022-247X(65)90032-6
[4] Brodskij, M. S.:Triangular and Jordan Representations of Linear Operators, Nauka, Moscow, 1969 (in Russian). English, transl.:Transl. Math. Monographs, vol 32, Amer. Math. Soc., Providence, R.I., 1971 [MR41#4283]. · Zbl 0224.20043
[5] Gel’fand, I. and Fomin, G.:Some Questions of Theory of Differential Equations, Nauka, Moscow, 1958 (in Russian).
[6] Gil’, M. I.: Estimating the norm of a function of a Hilbert-Schmidt operator,Soviet Math. 23(8) (1979), 13-19. · Zbl 0475.47016
[7] Gil’, M. I.: Estimation of the norm of the resolvent of a completely continuous operators,Math. Notes 26(5) (1979), 849-851. · Zbl 0443.47018
[8] Gil’, M. I.: On estimate for resolvents of nonselfadjoint operators ?near? to self-adjoint and to unitary ones,Math. Notes 33(2) (1983), 81-84. · doi:10.1007/BFb0068302
[9] Gil’, M. I.: On estimate for a stability domain of differential systems,Differential Equations,19(8) (1983), 1452-1454 (in Russian). · Zbl 0526.34039
[10] Gil’, M. I.: Absolute stability of nonlinear nonstationary systems with distributed parameters,Automat. Remote Control 46(6), part 1, (1985), 685-692. · Zbl 0583.93052
[11] Gil’, M. I.: Stability of essentially nonstationary systems,Soviet Phys. Dokl. 34(9) (1989), 753-755. · Zbl 0697.93050
[12] Gil’, M. I.: The freezing method for nonlinear equations,Differential Equations 25(8) (1989), 912-917. · Zbl 0714.34050
[13] Gil’, M. I.: Estimates for solutions of quasilinear parabolic systems,Differential Equations 25(4) (1989), 723-726 (in Russian). · Zbl 0706.35064
[14] Gil’, M. I.: Perturbation of the spectra of the certain class,Math. Notes 49(3) (1991), 328-330. · Zbl 0831.47008
[15] Gil’, M. I.: On estimate for norm of function of quasi-Hermitian operator,Studia Math. 103(1) (1992), 17-24. · Zbl 0812.47014
[16] Gil’, M. I.: On inequalities for eigenvalues of matrices,Linear Algebra Appl. 184 (1993), 201-206. · Zbl 0773.15007 · doi:10.1016/0024-3795(93)90379-3
[17] Gil’, M. I.: Estimates for norm of functions of matrices,Linear Multilinear Algebra 35 (1993) (to appear). · Zbl 0778.15015
[18] Gil’, M. I.: Operator function method in the theory of continuous systems, in: P. Borne and V Matrosov (eds),The Lyapunov Functions Method and Applications, J. C. Baltzer AG, Scientific Publishing Co. IMACS, 1990, pp. 69-71.
[19] Gohberg, I. C. and Krein, M. G.:Introduction to the Theory of Linear Nonselfadjoint Operators, Nauka, Moscow, 1965 (in Russian). English transl.:Transl. Math. Monographs. vol. 18, Amer. Math. Soc., R.I., 1969. · Zbl 0181.13503
[20] Gohberg, I. C. and Krein, M. G.:Theory and Applications of Volterra Operators in Hilbert Space, Nauka, Moscow, 1967 (in Russian). English transl.:Transl. Math. Monographs, vol. 24, Amer. Math. Soc., R.I., 1970 [MR 36-2007]. · Zbl 0194.43804
[21] Daletskii, Yu. and Krein, M.:Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R.I., 1974.
[22] Dunford, N. and Schwartz, J. T.:Linear Operators, part II: Spectral Theory, Selfadjoint Operators in Hilbert Space, Interscience, New York, London, 1963. · Zbl 0128.34803
[23] Elsner, L.: An optimal bound for the spectral variation of two matrices,Linear Algebra Appl. 71 (1985), 77-80. · Zbl 0583.15009 · doi:10.1016/0024-3795(85)90236-8
[24] Henry, D.:Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1981. · Zbl 0456.35001
[25] Kato, T.:Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1966. · Zbl 0148.12601
[26] Marcus, M. and Minc, H.:A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964. · Zbl 0126.02404
[27] Elayadi, Saber (ed.):Differential Equations. Stability and Control, Lecture Notes in Pure Appl. Math., vol. 127, Marcel Dekker, New York, Basel, Hong Kong, 1991.
[28] Stewart, G. W. and Sun, Ji-guang:Matrix Perturbation Theory, Academic Press, New York, 1990.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.