Suppose A ∈ L(Y ,Z ) , B ∈ L(X ,Y ) are Fredholm operators acting in linear spaces. By referring to the correspondence between Fredholm operators and their determinant systems, we derive the formulas for a determinant system for AB which are expressed via determinant systems for A and B. In our approach, applying results of the theory of determinant systems plays the crucial role and yields Cauchy-Binet type formulas. The formulas are utilized in many branches of applied science and engineering.
We consider continuous operators S +T in Banach spaces, where S is Fredholm and T is quasinuclear. By referring to the basic result of the Fredholm theory, i.e. to the expression of the resolvent ( I + λT ) −1 of the operator T as a quotient of entire functions of λ , we derive analogous formulas for generalized inverses of operators S +T. We apply the Plemelj-Smithies formulas describing terms of determinant systems for the quasinuclear perturbations of Fredholm operators.
In the paper we apply the modified powers of algebraic quasinuclei to construction of determinant systems for quasinuclear perturbations of Fredholm operators. Given two pairs (Ξ, X), (Ω, Y) of conjugate linear spaces, an algebraic quasinucleus F ∈ an (Ω → Ξ, X → Y) and a determinant system for the Fredholm operator S ∈op(Ω →Ξ, X → Y), we obtain algebraic formulas for terms of a determinant system for S + TF.
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