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On the sum of narrow and $C$-compact operators

N. M. Abasov^a, M. A. Pliev<sup>b

a</sup> MAI — Moscow Aviation Institute (National Research University), 3 Orshanskaya street, Moscow, 121552, Russia
^b Southern Mathematical Institute — the Affiliate of Vladikavkaz Science Center of the RAS, 22 Markus street, Vladikavkaz, 362027, Russia

Abstract: We consider narrow linear operators defined on a Banach–Kantorovich space and taking value in a Banach space. We prove that the sum $S+T$ of two operators is narrow whenever $S$ is a narrow operator and $T$ is a $(bo)$-continuous $C$-compact operator. For the proof of the main result we use the method of decomposition of an element of a lattice-normed space into a sum of disjoint fragments and an approximation of a $C$-compact operator by finite-rank operators.

Key words: Banach space, Banach–Kantorovich space, narrow operator, $(bo)$-continuous operator, $C$-compact operator.

UDC: 517.98

Received: 08.11.2017

DOI: 10.23671/VNC.2018.1.11391

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