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Let Δ T {\Delta _T} be the derivation on B ( H ) \mathfrak {B}(\mathcal {H}) defined by Δ T ( X ) = T X − X T ( T , X ∈ B ( H ) ) {\Delta _T}(X) = TX - XT(T,X \in \mathfrak {B}(\mathcal {H})) . We prove that if T T is an isometry or a normal operator, then the range of Δ T {\Delta _T} is orthogonal to the null space of Δ T {\Delta _T} . Also, we prove that if T T is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of Δ T {\Delta _T} is not all of B ( H ) \mathfrak {B}(\mathcal {H}) .

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

It is shown that the effective non-hermitean one-particle Hamiltonian Heff, deduced from many-body theory, is a measurable operator only if Heff is a normal operator; if Heff is not a normal operator, quasiparticles and the frequency-dependent quasiparticle-eigenfunctions are defined by the hermitean spectral-operator. Using perturbation-theory, an estimation of the frequency-dependent parts of the core-electron eigenfunctions is given which shows the limitations in defining a core-electron Hilbert subspace.

self adjoint operator A has a unique positive self adjoint square root A1/2; and an arbitrary self adjoint operator A has a unique representation A = A+ ~ A~ with A+ and A" self adjoint and positive and the range of each contained in the null space of the other. Write ( T\ = (T*Γ) 1/2 and, for complex λ and r ^ 0, let E(\, r) be the null space projection of (| T — XI \ — r/) +. For compact subsets K of the complex plane E(K)= Λ and for any Borel set M, E{M) = V {E(K): K compact and K £ M) . It is shown that E is the unique spectral measure such that T = \xE(dX) .

The purpose of this Chapter [J.] is to supply preparations for a new theory of functions F(N),where N is a normal operator. These functions are not ordinary compound functions, but they must be conceived as a suitable generalization of iterations of N,viz. N2 = N N, N3 = N N2 etc. Therefore the term “operational calculus” used by Stone (26) seems adequate.

Having discussed various aspects of the normal operator method on Riemann surfaces, and consequently on 2-dimensional Riemannian spaces, we now turn to their counterparts on higher dimensional Riemannian spaces. As far as functions, i.e. 0-forms, are concerned, it is plausible that all results thus far discussed, save those involving complex analyticity, can be reproduced for higher dimensions. That this is actually the case is shown in §1.

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.

Two applications of digital computers to control of crude oil distillation units are outlined. Results from both applications are quoted and comparisons made with normal operator control of the units. Difficulties encountered with the applications are mentioned and comments are made on steady-state control. The purpose of this paper is to describe two aspects of the application of digital computers to the control of crude oil distillation units.

If the normal operator has discontinuous spectrum, the multiplicity of an eigenvalue is easy to define. The case of continuous spectrum is more difficult, but the canonical multiplane-representation of the operator will be helpful. Our theory will define the multiplicity so that it will be not only precise, but intuitionally simple and even visual. We start with some hypotheses and preliminaries.

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

By Fuglede's theorem, an operator which commutes with a normal operator N must also commute with all spectral projections of N. We show that only in special cases is the theorem true when "near-commutativity" replaces "commutativity". We show too that the restriction of a normal operator to an almost reducing subspace may be far from all normal operators on the subspace. Introduction. One of the formulations of the Fuglede theorem is that if N is a normal operator on a Hilbert space and A is an operator such that NA=AN, then E(Y)A=AE(59) where E is the unique spectral measure associated with N and ,Y is an arbitrary Borel subset of the plane. It is not unreasonable to conjecture an asymptotic generalization of this fact. That is, if N is a normal operator, S9 a Borel subset of the plane, and A an operator such that INA-ANJI is small, then IIAE(Y)-E(Y)AII is small. Precisely, for every positive number s, there exists a positive number 6 such that j1NA-ANII

Theorem: every normal operator is a continuous function of a Hermitian one. Corollary: every normal operator on a separable Hilbert space is the sum of a diagonal operator and a compact one. THEOREM. Every normal operator is a continuous function of a Hermitian one. COROLLARY. Every normal operator on a separable Hilbert space is the sum of a diagonal operator and a compact one. Any two commutative Hermitian operators are Borel functions of some Hermitian operator; the version of the theorem in which "continuous" is replaced by "Borel" is old. Both the Borel and the continuous versions extend to countable sets of commutative Hermitian operators, with no additional effort. The present version is an easy consequence of well known Hilbert space techniques, together with an elegant fact of general topology. The result, with a similar but different proof, is implicit in a construction of Schwartz [6, pp. 15-16]. Much is known about normal operators, but not everything. The theorem is presented here in the hope that it may yield some new information. The basis of the hope is the derivation of the corollary from the theorem. The version of the corollary in which "normal" is replaced by "Hermitian" is the Weyl-von Neumann theorem; till quite recently the extension to the normal case was an open problem [4]. The first solution is due to I. D. Berg [1]. The proof below is shorter, and it may perhaps be considered more translucent. PROOF OF THE THEOREM. Every compact metric space is a continuous image of the Cantor set [5, ?41, VI]. If, in particular, A is a non-empty compact subset of the complex plane and 1F is the Cantor set in the unit interval, then there exists a continuous function p from 17 onto A. The elegant fact from general topology enters here: it is the existence of a Borel cross section. That is: there exists a Borel function Vp from A into F such that the composition p o Vy is the identity on A [5, ?43, IX]. (Short Received by the editors February 3, 1971. AMS 1970 subject classifications. Primary 47B15. I Research supported in part by a grant from the National Science Foundation. (American Mathematical Society 1972