In search of limit properties
“Concepts of convergence for linear operators and statements of stability” is the title of retired Chief Apostle Leber’s doctoral thesis. Read on for his full explanation.
This thesis is part of the domain of applied mathematics, more specifically, the field of functional analysis. The methods developed in this area make it possible to describe certain physical phenomena, particularly quantum mechanics.
The concept of the operator is fundamental to my work. An operator denotes an assignment of elements. Let’s take a simple example: we assign each number 1/n times its value, where n is a natural number: 1, 2, 3, etc. What arises from this is an (infinite) sequence of operators A1, A2, A3, etc. If you apply this sequence of operators to any given number, you will get a constantly decreasing sequence of numbers—or in mathematical terms: this sequence of operators “converges” to the operator A0, namely its multiplication by 0.
If you use general mathematical terms, the situation is not so simple. What does “convergence of operators” mean? In mathematical literature there are several definitions. In my paper I compared the various definitions with one another: which definition is narrower, which is broader?
The following question is also of interest: if a series of operators A1, A2, A3, … converge to an operator A0, and the operators A1, A2, A3, … all uniformly have a specific mathematical property, does the limit operator A0 also have this property (possibly even under special conditions)? In my work, I describe insights into this question as “statements of stability”.
Statements of stability of the aforementioned kind are not only of purely theoretical interest, but also have a certain practical use: at times it is difficult to prove a specific property of an operator A0. However, if one can construct a sequence of operators which are all known to have this property, and which converge to A0, then it is possible under certain circumstances that the limit operator A0 may also have this property.