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Maximal normal operators1 constitute a generalization of selfadjoint (hermitian) operations. Vagely speaking they may be conceived as operators with mutually orthogonal eigenvectors, but with complex eigenvalues. Selfadjoint operators and unitary operators are particular cases of maximal normal operators.

In this chapter the basic tools are created which will be used throughout the remainder of the book. The central topic is the Main Existence Theorem for principal functions, given in §1. The hypotheses of this theorem require the existence of normal operators. That such operators always exist is a nontrivial fact; its proof is given in §2 by constructing the operators L0 and L1 on an arbitrary Riemann surface. The method used there, which is typical of such problems, consists of constructing operators on compact bordered subregions and passing to a limit.

For A A an n n -normal operator on Hilbert space, we determine the irreducible representations of C ∗ ( A ) {C^ \ast }(A) , the C ∗ {C^ \ast } -algebra generated by A A and the identity. For A A a binormal operator, we determine an explicit description of the topology on the space of unitary equivalence classes of irreducible representations of C ∗ ( A ) {C^ \ast }(A) .

In this paper the following theorem will be proved: A necessary rind sufficient condition that the numerical range of a bounded normal )perator T in a Hilbert space is closed is that the set of extreme points ,f the convex closure of the spectrum of T contains no points of the coninuous spectrum of T. A few terms should be properly defined. The numerical range of a inear bounded operator T in a Hilbert space is defined as the set of ill complex numbers of the form (Tx, x) where x is a unit vector in She space. Clearly the numerical range of T contains the point spec-rum of T. Stone [2] proved that the numerical range of a bounded inear operator T is convex, and that if T is normal then the closure if its numerical range is exactly the convex closure K(T) of the ;pectrum of T. A point X of K(T) is defined as an extreme point of K(T) if no line segment joining any two points of K(T) other than X oontains X. Denote by A the set of extreme points of K(T). It is easy to see that A is contained in the spectrum of T, and that the convex losure of A is exactly K(T). Hence the sufficiency of the condition Follows readily from these facts. The necessity of the condition can be proved by contradiction. Suppose that there exists a point ao in A such that ao lies in the zontinuous spectrum of T. Now the numerical range of T is closed, and hence ao lies in it. So we can find a unit vector x in the space such that ao = (Tx, x). Since T is bounded and normal, it has a resolution of the identity {E,}, and T can be represented as an integral with respect to {EI}, [1], i.e.

In this paper we prove that the set of values of a normal-operator-valued function, defined and analytic on an open connected set in the complex plane, is commutative. The aim of this note is to prove that the set of values of a normaloperator-valued function, defined and analytic on an open connected set in the complex plane, is commutative. We were not able to find whether such a result is already known. Throughout the paper we denote by X the set of nonnegative integers and by L(X) the C -algebra of all bounded endomorphisms of a complex Hilbert space X. LEMMA. Let X be a complex Hilbert space and 9 the open unit disc in the complex plane. Let F: 9---L(X) be a normal-operator-valued analytic function given by (1) F(4) = AO + A14 + A2?42 + * ( e9). Then A.A. = A Ai (iiV6 ] jec //), AiA* = A*Ai (ieAl, j] V). PROOF. By the normality assumption we have (2) F(-)F(4) -F(4)*F( ) = 0 (c ?9). Since F is analytic on 2, the series

Whenever A = J. © A(\)p(d\) is a decomposable operator on a direct integral // = f. © H(\)p(d\) of Hilbert spaces and / is a function analytic on a neighborhood of o-(A), then we obtain that f{A(\)) is defined almost every- where and f(A)(X) = f(A(\)) almost everywhere. This relationship is used to study operators A, on a separable Hilbert space, for which some analytic func- tion A is a normal operator. Two main results are obtained. Let / be an ana- lytic function on a neighborhood of the spectrum of an operator A. If / (z) 4 0 for all z in the spectrum of A and if f(A) is a normal operator, then A is similar to a binormal operator. It is known that a binormal operator is unitarily equivalent to the direct sum of a normal and a two by two matrix of commuting normal opera- tors. As above if f(A) is normal and in addition, /(z) — £n has at most two roots counted to their multiplicity for each T in the spectrum of N, then A is a binor- mal operator. In this paper, all spaces are complex separable Hilbert spaces and all opera- tors on them are bounded. Various authors have investigated the structure of A whenever a particular function of A is normal. A. Brown (3) studied binormal oper- ators which included operators satisfying A = XI. He gave a structure theorem for binormal operators; these results have more recently been investigated in (2), and (15). J. Stampfli has shown that if A" is normal and A is invertible, then A is similar to a normal operator; subsequently, S. Foguel and C. Apostol have considerably generalized this result ((l), (lO), (12), (17)). In this paper, we generalize these results and show that it is possible to approach these and simi- lar problems through reduction theory of operators. Questions, concerning the struc- ture of operators, such as those investigated above will be reduced to questions about algebraic operators, which can then be handled with more elementary opera- tor theory methods. This approach seems new and essentially different from pre- vious works.