Partial Integral Operators on Banach–Kantorovich Spaces

Arziev A.D. Kudaybergenov K.K. Orinbaev P.R. Tangirbergen A.K.
August 2023 Pleiades Publishing

Mathematical Notes
2023 #114 Issue 1-2 15 - 29 pp.

Abstract: In this paper, we study partial integral operators on Banach–Kantorovich spaces over a ring of measurable functions. We obtain a decomposition of the cyclic modular spectrum of a bounded modular linear operator on a Banach–Kantorovich space in the form of a measurable bundle of spectra of bounded operators on Banach spaces. The classical Banach spaces with mixed norm are endowed with the structure of Banach–Kantorovich modules. We use such representations to show that every partial integral operator on a space with a mixed norm can be represented as a measurable bundle of integral operators. In particular, we show the cyclic compactness of such operators and, as an application, prove the Fredholm abla -alternative. We also give an example of a partial integral operator with a nonempty cyclically modular discrete spectrum, while its modular discrete spectrum is an empty set.

cyclically compact operator , measurable bundle of integral operators , modular spectrum , partial integral operator

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V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, 100170, Uzbekistan
Karakalpak State University named after Berdakh, Nukus, 230112, Uzbekistan
Regional Scientific and Educational Mathematical Center “North Caucasian Center for Mathematical Research” of the Vladikavkaz Scientific Center of Russian Academy of Sciences, Vladikavkaz, 363110, Russian Federation
K. Zhubanov Aktobe Regional State University, Aktobe, 030000, Kazakhstan

V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan
Karakalpak State University named after Berdakh
Regional Scientific and Educational Mathematical Center “North Caucasian Center for Mathematical Research” of the Vladikavkaz Scientific Center of Russian Academy of Sciences
K. Zhubanov Aktobe Regional State University

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