Extension Theory of Symmetric Second-Order Difference Operators

This thesis concerns itself with the extension theory of second-order difference equations in both a linear operator and linear relation framework. In Chapter 1, we introduce the extension theory of linear operators by means of both the von Neumann and the Krein-Vishik-Birman method. Chapter 2 is devoted to the construction of the non-negative, self-adjoint extensions of a particular class of second-order difference operators via the Krein-Vishik-Birman theory, with particular emphasis on the Friedrichs extension. We determine an explicit characterisation of such extensions, before applying this result to a second-order difference equation whose solutions are the Stieltjes-Wigert polynomials.Linear relations and their extensions are introduced in Chapter 3. In particular, a construction of the extremal maximal sectorial relations by Hassi et al. is considered. These results are utilised in Chapter 4 when we construct the extremal maximal sectorial extensions of the Discrete Laplacian with both the standard domain of square summable sequences and the sequences in that space with first component equal to 0.