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Title: Spectral bounds for infinite dimensional polydiagonal symmetric matrix operators on discrete spaces
Author: Sahovic, Arman
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2013
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In this thesis, we prove a variety of discrete Agmon Kolmogorov inequalities and apply them to prove Lieb Thirring inequalities for discrete Schrodinger operators on ℓ[superscript 2](ℤ). We generalise these results in two ways: Firstly, to higher order difference operators, leading to spectral bounds for Tri-, Penta- and Polydiagonal Jacobi-type matrix operators. Secondly, to ℓ[superscript 2]-spaces on higher dimensional domains, specifically on ℓ[superscript 2](ℤ[superscript 2]), ℓ[superscript 2](ℤ[superscript 3]) and finally ℓ[superscript 2](ℤ[superscript d]). In the Introduction we discuss previous work on Landau Kolmogorov inequalities on a variety of Banach Spaces, Lieb Thirring inequalities in ℓ[superscript 2](ℝ[superscript d]), and the use of Jacobi Matrices in relation to the discrete Schrodinger Operator. We additionally give our main results with some introduction to the notation at hand. Chapters 2, 3 and 4 follow a similar structure. We first introduce the relevant difference operators and examine their properties. We then move on to prove the Agmon Kolmogorov and Generalised Sobolev inequalities over ℤ of order 1, 2 and σ respectively. Furthermore, we prove the Lieb Thirring inequality for the respective discrete Schrodinger-type operators, which we subsequently lift to arbitrary moments. Finally we apply this inequality to obtain spectral bounds for tri-, penta- and polydiagonal matrices. In Chapter 5, we prove a variety of Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript 2]) and ℓ[superscript 2](ℤ[superscript 3]). We use these intuitive ideas to obtain 2[superscript d-1] Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript d]). We continue from here in the same manner as before and prove the discrete Generalised Sobolev and Lieb Thirring inequalities for a variety of exponent combinations on ℓ[superscript 2](ℤ[superscript d]).
Supervisor: Laptev, Ari Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available