Unbounded Normal Operators Research Articles

Unbounded Normal Operators in Hilbert Space

Unbounded Normal Operators in Hilbert Space

Unbounded normal operators in Hilbert space

The theory of abstract unbounded operators and their extensions developed along two main lines. The first is the theory of extensions of symmetric operators (Achiezer and Glasman [3]). In this case the problem is to extend a symmetric operator to a self-adjoint one. This theory includes necessary and sufficient conditions for the existence of such extensions, as well as complete characterization of the nature of these extensions, especially the nature of the spectrum of various extensions pertaining to the same operator. This theory imposes severe conditions and can be applied only in particular cases. In order to deal with abstract problems arising in the theory of partial differential equations and integral equations a relaxation is needed. The notions of generalized solution, solution, etc. (for example in Lax [1]) and the theory of distributions lead to the following problem of extension of general abstract operators. (For the sake of simplicity the illustration below will be confined to operators acting in Hilbert space.) The operator (differential, integral, etc.) !T is embedded in a certain space E. In this space a domain Do is considered. Denote Y acting on Do by To. To is in general bounded below on Do so it may be closed to an operator To ' To acting on Do D bo. By this process the generalized are obtained. In most cases the range of To is not the whole space so an extension is needed. Among all possible extensions only the are considered. They are obtained by looking at a natural formal adjoint 9* defined on the closure of the set bo. Denote this operator by T* and denote (T1*)* by T1. Any function in the domain of T1 is called a solution. In general the range of T1 is the whole space but there are too many weak solutions because there are null solutions. So the problem is to find an intermediate To c T c T1 having a bounded inverse defined on the whole space. Somewhat different versions are in G. C. Rota [4] and Gohberg and Krein [5]. These general theories have only weak results, because of the generality of the operators considered.

Spectral representations for some unbounded normal operators

Introduction. In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (U,,, = U,LUV) as well as many others. In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures. We shall consider therefore a mapping z -* UZ of a locally compact space Z into the set of unbounded normal operators on a Hilbert space. Since the operators are not necessarily bounded, the conditions imposed on the mapping z -* Uz vary somewhat from the conditions in the bounded case. The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8]. The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea. For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21]. Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A. E. Nussbaum [19]. Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]). The setting of the problem in algebras of measures is given in ??1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in ??3 and 4. The main result is stated and proved in ?6. In ?8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].

Spectral Representations for Some Unbounded Normal Operators

Spectral Representations for Some Unbounded Normal Operators

On the Permutability of Normal Operators

1. In this paper we obtain a theorem about the permutability of (in general unbounded) normal operators. This theorem is analogous to a theorem proved by the present authors in collaboration with J. von Neumann [3] concerning the permutability of self-adjoint operators. As we shall point out later it is not possible to generalize precisely the conditions given in [3] to the case of normal operators. Nevertheless, the proof of the theorem given here, with essential modifications, follows the proof given in [3]. We give our theorem and proof in Section 2. In Section 3 we apply this theorem on the permutability of normal operators to obtain the integral representation for a semi-group {N,} of normal operators (in general unbounded), where the index x ranges over a semi-group which is a cartesian product I+ X E where I+ is the set of vectors in m-dimensional euclidean space with non-negative integer components and Et is the set of vectors in n-dimensional euclidean space with non-negative components. This theorem contains as special cases theorems by E. Hille [6], B. v.Sz. Nagy [9, pp. 74-76], A. Devinatz [2] and a theorem recently published by R. Getoor [5]. It should be pointed out that our theorem represents an essential generalization of these previous results. For the case where { Nx ; x > 0 } is a one parameter semi-group of unbounded normal operators, the method of Nagy [9, pp. 74-75] may be pushed through to obtain the integral representation, and only in this case. However, with the use of our permutability theorem we are able to materially simplify the proof for the one parameter case. Moreover, in the unbounded case, even with the use of our permutability theorem, there remain some difficulties for multi-parameter semi-groups which do not allow us to use immediately the theorem for the one-parameter case. If the index x ranges over a more general Abelian topological semi-group than we consider, the representation problem is still open, although for semi-groups of normal operators which are uniformly bounded a method introduced by R. Phillips [8] may be used (cf. also A. E. Nussbaum [7]). 2. We start with the definition of permutability of two (in general unbounded) normal operators defined in a Hilbert space & (cf. Nagy [9] p. 50). DEFINITION 1. Let N1 and N2 be normal operators and {K1(z)} and {K2(Z)I their corresponding canonical resolutions of the identity. We say that N1 and N2 permute if Kl(zi)K2(z2) = K2(z2)Kl(zi) for each complex z1 and z2 It should be pointed out here that this definition is, by a theorem of B. Fuglede [4], equivalent to the usual definition of permutability in the case where N1 and N2 are bounded (N1N2 = N2N1) or in the case where N1 is bounded and N2 is unbounded (N1N2 _ N2N1). DEFINITION 2. We say that the operators N1, N2 and N have the property P if they are normal operators and if N = N1N2 = N2N1 .

On Semi-Groups of Unbounded Normal Operators

On Semi-Groups of Unbounded Normal Operators

On semi-groups of unbounded normal operators

On semi-groups of unbounded normal operators