Geodesic
Metric properties of the Rayleigh--Ritz operator
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2022), pp. 54-63

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We consider a nonlinear functional Rayleigh–Ritz operator defined on a set of pairs of measurable functions and 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.
Keywords: space of measurable functions, convergence by measure, non-invariant metric, Rayleigh–Ritz operator. 
@article{IVM_2022_9_a4,
     author = {A.V.Lakeev and Yu. \`E Linke and V. A. Rusanov},
     title = {Metric properties of the {Rayleigh--Ritz} operator},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {54--63},
     publisher = {mathdoc},
     number = {9},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_9_a4/}
}
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A.V.Lakeev; Yu. È Linke; V. A. Rusanov. Metric properties of the Rayleigh--Ritz operator. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2022), pp. 54-63. http://geodesic.mathdoc.fr/item/IVM_2022_9_a4/