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P-Fredholmness of Band-dominated Operators, and its Equivalence to Invertibility of Limit Operators and the Uniform Boundedness of Their Inverses

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Title: P-Fredholmness of Band-dominated Operators, and its Equivalence to Invertibility of Limit Operators and the Uniform Boundedness of Their Inverses
Author(s): Moilanen, Eero
Contributor: University of Helsinki, Faculty of Science
Degree program: Master 's Programme in Mathematics and Statistics
Specialisation: Mathematics
Language: English
Acceptance year: 2022
Abstract:
In the thesis ”P-Fredholmness of Band-dominated Operators, and its Equivalence to Invertibility of Limit Operators and the Uniform Boundedness of Their Inverses”, we present the generalization of the classical Fredholm-Riesz theory with respect to a sequence of approximating projections on direct sums of spaces. The thesis is a progessive introduction to understanding and proving the core result in the generalized Fredholm-Riesz theory, which is stated in the title. The stated equivalence has been further improved and it can be generalized further by omitting either the initial condition of richness of the operator or the uniform boundedness criterion. Our focal point is on the elementary form of this result. We lay the groundwork for the classical Fredholm-Riesz theory by introducing compact operators and defining Fredholmness as invertibility on modulo compact operators. Thereafter we introduce the concept of approximating projections in infinite direct sums of Banach spaces, that is we operate continuous operators with a sequence of projections which approach the identity operator in the limit and examine whether we have convergence in the norm sense. This method yields us a way to define P-compactness, P-strong converngence and finally PFredholmness. We introduce the notion of limit operators operators by first shifting, then operating and then shifting back an operator with respect to an element in a sequence and afterwards investigating what happens in the P-strong limit of this sequence. Furthermore we define band-dominated operators as uniform limits of linear combinations of simple multiplication and shift operators. In this subspace of operators we prove that indeed for rich operators the core result holds true.


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