Estimates for norms of matrix-valued and operator-valued functions and some of their applications

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.