Metric Properties of the Rayleigh–Ritz Operator

We consider a nonlinear functional Rayleigh–Ritz operator that is defined on a set of pairs of measurable functions and is equal to the ratio of their modules if the denominator is nonzero and zero otherwise. We investigate the continuity of this operator with respect to the convergence of the measure. It is shown that the convergence of the operator value on the sequence of pairs to the value on the limit pair of functions requires not only convergence as its arguments, but also convergence as the carriers of the second argument to the carrier of its limit. The results obtained have applications in the theory of differential realization (in Hilbert space) of higher-order nonlinear dynamic models.