Introduction to Operator Theory (Essay I)

Course Introduction

This course is based primarily on John B. Conway’s “A Course in Operator Theory” book. The goal is to develop a rigorous foundation suitable for graduate-level pure mathematics, while presenting the material in a way that remains meaningful to students in the applied sciences.

Operator Theory sits naturally at the intersection of analysis, physics, and modern machine learning. While such applications will not be assumed or developed initially, the theory presented here is the mathematical framework on which many contemporary results are built (for further reading, see the footnotes).

Again, this is so you understand there exist many applications of this theory, and I will make no assumptions that you initially understand what a $C^{*}$ -Algebra is or Banach algebra, or even an algebra is for that matter. Instead, I will tailor my explanations, assuming a strong background in Calculus and Linear Algebra, and some understanding of Topology.

What is an Operator?

Operators are a broad concept in mathematical literature, and are often associated with functions that map elements from one space to another. However in this course, an operator will always mean a linear operator acting between topological vector spaces.

Recall, for any two vector spaces $X$ and $Y$ over the field $\mathbb{F}$ , an operator $\mathbf{S}: X \longrightarrow Y$ is linear, provided it satisfies

$\displaystyle \mathbf{S}(\lambda u + \phi v) = \lambda \mathbf{S}(u) + \phi \mathbf{S}(v)$

for $\lambda, \phi \in \mathbb{F}$ and $u, v \in X$ .

Example 1.

A trivial example of this would be any square matrix $A \in \mathcal{M}_{n}(\mathbb{R})$ , operating on vectors in $\mathbb{R}^{n}$ . Namely we construct the mapping $\mathbf{T}_{A}: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n}$ , defined by

$\displaystyle \mathbf{T}_{A}(x) = Ax$

For all $x \in \mathbb{R}^{n}$ . The linearity of $\mathbf{T}_{A}$ is often shown in a standard Linear Algebra course. Yet this example should be viewed as a finite-dimensional prototype of a much broader theory. Now lets provide another example of linear operators.

Example 2.

Let $L^{2}[0, 1]$ be the set of square-integrable functions on the interval $[0,1]$ . Namely for any complex-valued, measurable function $f \in L^{2}[0,1]$ , it is necessarily the case

$\displaystyle \int \limits_{0}^{1} |f(t)|^{2} dt < \infty$

Then for some $\tau \in \mathbb{R}$ , we introduce the Volterra operator $\mathbf{V}: L^{2}[0, 1] \longrightarrow L^{2}[0, 1]$ , defined by

$\displaystyle (\mathbf{V}g)(\tau) := \int \limits_{0}^{\tau}g(t) dt$

which takes $g \mapsto \mathbf{V}g$ for any $g \in L^{2}[0, 1]$ . This is clearly linear, since

$\displaystyle \mathbf{V}(\lambda f + \omega g) = \int \limits_{0}^{\tau}(\lambda f + \omega g) dt$

$\displaystyle = \lambda \int \limits_{0}^{\tau}f(t)dt + \omega \int \limits_{0}^{\tau}g(t)dt$

$\displaystyle = \lambda (\mathbf{V}f)(\tau) + \omega (\mathbf{V}g)(\tau)$

This example was purposefully chosen, since historically functional analysis was crafted by the Italian mathematician and physicist, Vito Volterra, who, in his work dealing with integral equations, began to interpret functions as vectors defined on infinite-dimensional vector spaces.

Inner Products

Although you likely learned about inner products from Linear Algebra, we will revisit them briefly. Particularly in $\mathbb{R}^{n}$ , the inner products is shown:

$\displaystyle <x, y> := \sum \limits_{i = 1}^{n} x_{i}y_{i}$

to which your professor moved on to prove how this product induces a norm, or can be used for interpreting angular relationships between vectors. Another example is often provided:

$\displaystyle <u, v> := \int \limits_{a}^{b} u(t) \overline{v(t)} dt$

for any two functions $u, v \in C([a, b])$ where $C([a, b])$ is the space of continuous complex-valued functions. This example induces an inner product whose completion is the space of square-integrable functions in $[a, b]$ . However when done correctly, the student shouldn’t miss the underlying connection here. The inner product $<x, y>$ is described in $\mathbb{R}^{n}$ , which is a finite-dimensional vector space, but $<u, v>$ is what we consider once we ask about the corresponding inner product in an uncountably infinite number of dimensions. That is because the integral can be seen as an infinite dimensional analogue of the sum.

So now that we understand one can view the space of continuous functions as an infinite dimensional vector space, we should ask about operators over these spaces. Linear algebra commonly treats operators over spaces of finite dimension (commonly referred to as matrices), but what we discussed implies linear algebra needs more generality. This is exactly the study of operator theory.

Click here for part II: On the Hierarchy of Topological Spaces

Further reading:

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Introduction to Operator Theory (Essay I)

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