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.
Highlights
Under Condition IV, a solution of System (8) can be represented as the sum of a Laurent series with a pole of order 1 at origin
When operator A22 is invertible, the problem can be reduced to solution of linear equation
A21 is sufficiently small, the equation can be written as A11 x1 − λBx1 = y1 − A12 A22
Summary
The Frobenius formula is powerful tool in contemporary applications of linear algebra [1]. The construction of the approximate methods in bifurcation theory [16] in the case of block operators in Banach spaces employed the Frobenius formula. System (1), as in the case of the classical Frobenius formula for block matrices, can be rewritten as two equations with two unknown vectors, which are elements of different functional Banach spaces. Such parameters can be found from the following system of linear algebraic equations constructed after substitution of x2 by virtue of the solvability condition (2), i.e., from system. Let conditions of Theorem 1 be fulfilled, det B 6= 0; operator A is continuously invertible, ξ = B−1 d and using Formulae (4) and (6), operator A−1 can be explicitly constructed in block form. From Theorem 1 it follows generalisation of classic Frobenius formula on the case of block operators in normed spaces.
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