Documents

A note on the definition of an unbounded normal operator
In the literature, one sees two definitions for what it means for a densely defined operator to be normal. There is no harm in having these competing definitions as they are, in fact, equivalent. While the equivalence of these statements seems to be widely known, the proof is not obvious to me, and my efforts to find a detailed and self-contained proof of this fact have left me empty-handed. I give such a proof here.
An example of a non-separable Hilbert space
A note on how to construct a non-separable Hilbert space.
Dissertation: Pettis integration with applications to quantum Markov semigroups
In the most general circumstances, the generators of quantum Markov semigroups are unbounded. In this dissertation we investigate the form of such generators. Along the way, we realize that a more general concept of Pettis integration is needed to discuss whether or not the generators are closed and we develop tools to help with this discussion.
The Regularity Theorem for distributions
The purpose of these notes is to give a proof of the one-dimensional Regularity Theorem for distributions which states that if T is a tempered distribution on the real line then T is the weak nth derivative of some polynomially bounded continuous function. The notes start by giving a proof of the N-Representation Theorem for the Schwartz class and for tempered distributions.
Riesz bases and unconditional bases
In this note we give a brief introduction to adjoint operators on Hilbert spaces and a characterization of the dual of a Hilbert space. We then introduce the notion of a Riesz basis and give some equivalent definitions. Afterwards, we discuss unconditional bases and prove that the Haar system is an unconditional basis for L_p(R), for p>1, and that L_1[0,1] does not contain an unconditional basis.
The vector-valued maximal function
In this note we define a vector-valued version of the Hardy-Littlewood Maximal function and then prove that it is weak-(1,1) and strong-(p,p), for all p>1. The note starts by giving a proof of the Calderon-Zygmund Lemma.
Interpolation of Operators: Riesz-Thorin and Marcinkiewicz
The purpose of this note is to give detailed proofs of the interpolation theorems by Riesz-Thorin and Marcinkiewicz as well applications for both. It starts by giving a proof of Hadamard’s Three-Line Theorem.
Master’s Thesis: Solutions to Pell Equations
In the classical sense, a Pell equation is a Diophantine equation of the form x^2-dy^2=1, where d is not a perfect square. The problem of finding integer solutions to such equations was first proposed by Fermat in the 17th century but its significance has reverberated well into the 20th. Most notably, Pell equations helped to solve Hilbert’s tenth problem. This thesis gives the solution to all Pell equations using infinite continued fractions.