# Ph.D. thesis

**Author's comments:** There are two, related parts to this thesis: one on representations of Lie semi-groups and one on operators associated to representations of Lie groups. The first part was published in the Canadian Journal of Mathematics, but the second was published only as an announcement in the Proceedings of the National Academy of Sciences of the USA. It nevertheless had the good fortune to be taken seriously by Derek Robinson, who incorporated some of the results into his book on **Elliptic Operators and Lie Groups**.

*Examining again, after forty years, the verification of the basic estimates of the second part, I found a large number of misprints, so that it would have been difficult if not impossible to follow my arguments line by line. I have tried in the present version to correct the misprints, but cannot be certain to have fully succeeded. The notation sometimes takes a few minutes to decipher but appears to be comprehensible.*

*The thesis remains, to my regret, my only active encounter with partial differential equations, a subject to which I had always hoped to return but in a different vein.*

**Added April, 2017.** This thesis was written by me, with only a formal advisor, in 1959 and typed by my wife, Charlotte. I have examined it again, but find a good number of lines and formulas impossible to understand precisely. I would advise anyone who wishes to examine the material, at least the material on holomorphic semi-groups, to consult the book of Derek Robinson, in which there are several precise references to the thesis:

Elliptic operators and Lie groups. *Clarendon Press, Oxford u. a. 1991, ISBN 0-19-853591-0.*

**Added January, 2018.**** **The author and the editor are grateful to Derek Robinson for two very important contributions to this section of the site. The first was a list of the many misprints and/or small errors in the original text. The second is a clear, although brief, description of the context in which the thesis was written, or rather the context in which it is best to consider it. I add a few autobiographical remarks.

*My graduate education was in two stages: a year at UBC, for a master's degree was followed by two years at Yale for a doctor's degree. In neither case was my formal supervisor anything more than formal. At UBC, the thesis was on commutative algebra. The topic I chose myself, but after a seminar with Prof. Douglas Murdoch from Northcott's brief text Ideal Theory. The thesis was undoubtedly not well-written and could be understood by no-one. Moreover, I myself discovered very soon after submission an error in the arguments. It was nevertheless decided to award me the degree, presumably so that I could profit from my successful application to Yale. This was generous of the mathematics faculty at UBC and, of course, decisive for my life.*

*At that time, functional analysis was the principal topic at Yale, the principal texts being the first volume of Dunford-Schwartz and the tome Functional Analysis and Semi-Groups by Hille-Phillips. Hille was trained as a classical analyst and their book, certainly thick, was, in spite of the title, a rich introduction to many topics of classical analysis. I spent a good deal of time with it, but also with many of the paper-back reprints available at the time. Analysis as such was represented by Felix Browder, whose course on partial differential equations was given extemporaneously, although seldom clearly, but none the less instructively. I spent a good deal of time transforming his repeated attempts, not always successful, to reach an end with the proof of this or that assertion into lecture notes that I could understand. There are traces of all this in my thesis, written, I think I can assert truthfully, quite independently. Once again, there was, oddly enough, no-one to understand it, but as I know from a conversation overheard in a stairway, Browder was quite firm in defending me and my thesis in the face of another faculty member, whose stated grounds for rejecting it were solely that no-one could read it.*

*Although, by the end of my time at Yale, I had already, as a result of listening to lectures by Steven Gaal on automorphic forms and the work of Selberg, begun to think of other things, I had become an analyst, even a functional analyst, and, I believe I can assert, have never stopped being one. This has many advantages, even in the context of more classical branches of our science, especially that of independence.*