Abstract
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 result 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.
Original language | English |
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Pages (from-to) | 59-88 |
Number of pages | 30 |
Journal | Acta Applicandae Mathematicae |
Volume | 32 |
Issue number | 1 |
DOIs | |
State | Published - 1 Jul 1993 |
Keywords
- Mathematics subject classifications (1991): 47A56, 47A55, 15A54, 35A30, 34A30
- Operator-valued functions
- eigenvalues
- matrices
- nonself-adjoint operators
- spectrum perturbation
- stability
ASJC Scopus subject areas
- Applied Mathematics