On the linear independence of the Keldysh derived chains for operator-valued functions analytic in a~half-plane

G. V. Radzievskii

Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 547-567

Résumé

Criteria are established for linear independence of Keldysh derived chains constructed from the root vectors of functions analytic in the left half-plane with values in the set of operators acting in a Hilbert space $\mathfrak H$. In particular, an operator-valued function $L(\lambda)=L_0+\lambda L_1+\dots+\lambda^nL_n$ is considered. Let $\operatorname{Im}L(i\tau)\geqslant0$ for $\tau\in\mathbf R$ and suppose that zero does not belong to the numerical range of the operator $L(i\tau_0)$ for some $\tau_0\in\mathbf R$. Denote by $x_\mu$ an eigenvector $L(\tau)$ corresponding to an eigenvalue $\mu$, and by $M$ the subset of eigenvalues $\mu$ for which $\operatorname{Re}\mu0$ and $i(L'(\mu)x_\mu,x_\mu)0$ for $\operatorname{Re}\mu=0$. Then it is proved that the vectors $\widetilde y_\mu=\{x_\mu,\mu x_\mu,\dots,\mu^{m-1}x_\mu\}$ that belong to the direct sum of $m$ copies of the space $\mathfrak H$ are linearly independent when $\mu\in M$ while $m\geqslant[(n+1)/2]$. If, moreover, the operator $(i)^nL_n\geqslant0$, then this assertion holds also for $m=[n/2]$. A connection is exhibited between the results obtained here and the question of uniqueness of the solution of a problem on the half-line for systems of ordinary differential equations with constant coefficients. Bibliography: 7 titles.