## Functional Analysis and Semi-groups, Volume 31, Part 1Early in 1952 it became obvious that a new printing would be needed, and new advances in the theory called for extensive revision. It has been completely rewritten, mostly by Phillips, and much has been added while keeping the existing framework. Thus, the algebraic tools play a major role, and are introduced early, leading to a more satisfactory operational calculus and spectral theory. The Laplace-Stieltjes transform methods, used by Hille, have not been replaced but rather supplemented by the new tools. - Foreword. |

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According addition algebra analytic applies argument assertion assume assumption belongs bounded called Chapter clear closed commutative compact complex condition Consequently consider contains continuous converges corollary corresponding defined DEFINITION denote dense differentiable domain element equal equation exists extension fact Finally finite fixed follows formula function Further given gives hand hence holds holomorphic ideal implies infinitesimal integral interval Lemma limit linear bounded mapping maximal measurable norm obtain obvious operator operator topology particular positive problem PROOF properties prove References regular relation representation resolvent respect result satisfies semi-group semi-group of class sense sequence shows ſº solution space spectrum strong strongly subadditive subset sufficient Suppose Theorem theory transformation uniform uniformly unique unit element values vector weakly zero