# 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.*