In the pioneering paper [Commun. Pure Appl. Math. 21, 467-490 (1968; Zbl 0162.411)], \textit{P. Lax} stated a condition under which certain one parameter families of operators \(\{\) L(t)\(\}\) are isospectral, i.e., all the L(t) have the same spectrum. Lax acutally asserted that his condition implied the unitary equivalence of all the L(t). Unfortunately it is not clear from the Lax's paper exactly when this condition is applicable. Furthermore there seems to be no clear statement nor proof of this in the literature [see \textit{W. Eckhaus} and \textit{A. van Harten}, The inverse scattering transformation and the theory of solitons. An introduction (1981; Zbl 0463.35001), Chapter 3 where this is discussed]. The purpose of this note is to state and prove such a result and show as Lax affirmed that it can indeed be used to prove the unitary equivalence of Schrödinger operators whose potentials evolve according to the Korteweg-de Vries equation. The main theorem depends heavily on \textit{T. Kato}'s paper [J. Fac. Sci. Univ. Tokyo, Sect. IA 17, 241-258 (1970; Zbl 0222.47011)]. We refer to it for background and notation.