CHARTING THE OPERATOR TERRAIN INTRODUCTION The territory of bounded linear operators on a Hilbert space is vast. There are basically two regions of civilization (the normal operators and the compact operators) which are surrounded by various more recent settlements. Beyond this frontier there remains an enormous, aljnost completely virgin, wilderness. The purpose of this memoir is to offer a cartographic procedure for bringing some organizational sense to the prodigious task of exploring this varied operator terrain. While we will prove (in a certain sense) that the classification of all operators up to unitary equivalence is an essentially unattainable objective, we hope this theory will be serviceable in colonizing some additional enclaves, as well as suggesting other more rugged areas which one might probe in search of some fascinating and unusual phenomena. Our cartographic system is essentially an adaptation to single operators of John von Neumann's classification and reduction theory, in which the study of general rings of operators is reduced to the study of certain minimal building blocks called factors. Our single operator version is cast into a mold which exhibits it as a natural generalization of the spectral theorem and the spectral multiplicity theory for normal operators. To each bounded opera- tor we associate a certain set of (equivalence classes of) "factor operators," which we call the quasi-spectrum of the operator. This object may be identi- fied with the ordinary spectrum of the operator precisely when the operator is normal. An arbitrary bounded operator is determined, up to unitary equiva- lence, by a triplet consisting of its quasi-spectrum, a finite measure class on that quasi-spectrum, and a "spectral multiplicity function" defined on the measure classes absolutely continuous with respect to the first mentioned measure class. In a certain imprecise and audacious sense, this reduces the unitary equivalence problem for general operators to a (weaker) equivalence problem for "factor operators." The spectral theorem and the spectral multi- plicity theory for normal operators are (very) special cases of this general theory. About thirty years ago Gelfand and Naimark generalized the spectral theorem for normal operators to a theorem about commutative C -algebras. Since that time a beautiful, deep and extensive theory for noncommutative C - algebras has developed, as an examination of Jacque Dixmier's basic treatise Received by the Editors December 6, 1973, and in revised form December l6, 1975. 1
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