Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces

Nikolai A. Sidorov, Aliona I. Dreglea and Denis N. Sidorov
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Nikolai A. Sidorov: Institute of Mathematics and Information Technologies, Irkutsk State University, 664025 Irkutsk, Russia
Aliona I. Dreglea: Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia
Denis N. Sidorov: Institute of Mathematics and Information Technologies, Irkutsk State University, 664025 Irkutsk, Russia

Mathematics, 2021, vol. 9, issue 23, 1-9

Abstract: The efficient construction and employment of block operators are vital for contemporary computing, playing an essential role in various applications. In this paper, we prove a generalisation of the Frobenius formula in the setting of the theory of block operators on normed spaces. A system of linear equations with the block operator acting in Banach spaces is considered. Existence theorems are proved, and asymptotic approximations of solutions in regular and irregular cases are constructed. In the latter case, the solution is constructed in the form of a Laurent series. The theoretical approach is illustrated with an example, the construction of solutions for a block equation leading to a method of solving some linear integrodifferential system.

Keywords: Frobenius formula; operator theory; block operators; Fredholm operator; asymptotic approximations; Nekrasov–Nazarov method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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