The abstract Fatou theorem and the signal transmission on Thomson cables

Abstract

The Fatou theorem on the Poisson representation of bounded harmonic functions on a half space is generalized to the bounded solutions $u(t)$ of the second order equation $$ u''(t) = A u(t), 0 < t < \infty, $$ in a dual Banach space $X = X_*{'}$, when $A$ is the dual of a non-negative operator $A_*$ with dense domain in $X_*$. Any bounded weak* solution is represented as $u(t) =$ $\exp(-t\sqrt{A})f$ with the weak* initial value $f$. Its prototype is in A.~V. Balakrishnan's paper in 1960 on fractional powers of non-negative operators. This is applied to prove the uniqueness of solutions in the theory of signal transmission on submarine cables by W. Thomson in 1855.