Norm estimates for functions of matrices with simple spectrum
In 1958 \textit{I. M. Gelfand} and \textit{G. E. Schilow} [Some Questions of Theory of Differential Equations. Moscow: Nauka (1958); cf. Einige Fragen zur Theorie der Differentialgleichungen. Berlin: VEB Deutscher Verlag der Wissenschaften. 195 S. (1964; Zbl 0133.37503)] established an estimate for the norm of a~regular matrix-valued function in connection with their investigations of partial differential equations. However, the estimated obtained was not sharp. In the present paper the author establishes sharp estimates for the norm of matrix-valued functions of a~matrix having geometrically simple eigenvalues~\((\mu_n)\). The regularity of the function in question is not assumed. The main result is as follows: Let all the eigenvalues of an~\(n\times n\) complex matrix be geometrically simple, and let \(f\)~be a~function defined on~\(\sigma(A)\). Then \[ \|f(A)-f(\mu_m)I\|\leqslant \gamma_0 \sum^{m-1}_{k=1} |f(\mu_k)-f(\mu_{k+1})|, \] where \(\gamma_0= 1+\max_{j\leq m} l(L_j) p(A,\rho_j)\). If in addition the condition \(f(\mu_{k+1})\leqslant f(\mu_k)\), \(k=1,\dotsc,m-1)\), holds, then \(\|f(A)-f(\mu_m)I\|\leq \gamma_0|f(\mu_1) - f(\mu_m)|\). Here \(l(L_j)=(2\pi)^{-1} \int_{L_j}|dz|\), \(L_j\) is a~Jordan closed contour surrounding the eigenvalues \(\mu_1,\dots,\mu_j\), \(p(A,x)=\sum^{n-1}_{k=1} g^k(A)/[x^{k+1}\sqrt{k!}]\), \(x>0\), and \(\rho_j=\min\{|\lambda-\mu_k|\mid \lambda\in L_j\), \(k=1,\dotsc,j\}\). Also, applications of the estimates obtained to the periodic problem and two-point boundary value problem for vector differential equations are discussed.
scientific article; zbMATH DE number 5800238