Showing papers in "Pacific Journal of Mathematics in 1961"

Operational calculus of linear relations

Abstract: (unbounded spectrum, domain Φ X) in finite-dimens ional spaces which linear operators exhibit only in infinite-dime nsional spaces. We present an outline of the paper. In § 2 we define p(T) where p is a polynomial with coefficients in the field Φ involved in X. We prove that (pq)(T) = p(T)q(T), (poq)(T) = p{q{T)), and point out that sometimes (p + q){T) Φ p(T) + q(T), etc. In § 3 we turn to relations in dual pairs. In this situation, adjoints can be defined. We build an automorphism λ —> λ of Φ into the theory of dual pairs, so as not to exclude the Hubert space situation, which dual pairs are intended to imitate. (Thus the transpose is a special kind of adjoint.) Closedness is defined algebraically, but in a way compatible with the topological concept. Closure of T 7* and other algebraic properties of * are established. Finally, it is shown that if T is closed and its resolvent is not void then p(T) is also closed. Section 4 considers the self-dual case. We give a simple condition (4.3) always true in Hubert space, that T*T be self-adjoint, T being closed. In § 5 we give the spectral analysis of self-ad joint linear relations in Hubert space. In a 1:1 manner these correspond to the unitary operators, via the Cay ley transform. However, it can be shown directly that X is the direct sum of orthogonal subspaces Y, Z which reduce T (= T7*) giving in Za self-ad joint operator and in Fthe inverse of the zero-operator. 2 Linear relations* A relation T between members of a set X and members of a set Y is merely a subset of X x Y. For x e X, T(x) = {y (x, y) e T}. The domain of T consists of those x such that T(x) is not void. T is called single-valued if T(x) never contains more than one element. The range of T is the union of all T(x).

A generalization of the Stone-Weierstrass theorem.

Abstract: 1# Introduction. Consider a compact Hausdorff space X and the set C(X) of all continuous complex-valued functions on X, Consider also a subset 21 of C(X) which is an algebra, which is closed in the uniform topology of C(X), which contains the constant functions, and which contains sufficiently many functions to distinguish points of X. Such an algebra 21 is called self-adjoint if the complex conjugate of each function in 2t is in 21. The classical Stone-Weierstrass Theorem states that if 21 is self-ad joint then 21 = C(X). If 21 has the property that the only functions in 21 which are real at every point of X are the constant functions then 21 is called anti-symmetric. Clearly anti­ symmetry and self-adjointness are opposite properties, in the sense that if 21 has both properties then X must consist of a single point. Hoffman and Singer [2] have studied these two properties and given several interesting examples. The present paper was inspired by their work but it more directly relates to a previous paper of Silov [3]. The purpose of the present paper is to prove the following decomposition theorem for a general algebra 21 of the type defined above.

Axioms for non-relativistic quantum mechanics.

Abstract: In the approach to the axiomatization of quantum mechanics of George W. Mackey [7], a series of plausible axioms is completed by a final axiom that is more or less ad hoc. This axiom states that a certain partially ordered set — the set P of all two-valued observables — is isomorphic to the lattice of all closed subspaces of Hilbert space. The question arises as to whether this axiom can be deduced from others of a more a priori nature, or, more generally, whether the lattice of closed subspaces of Hilbert space can be characterized in a physically meaningful way. Our central result is a characterization of this lattice which may serve as a step in the indicated direction, although there is not now a precise sense in which our axioms are more plausible than his. Its principal features may be described as follows.

GROUPS WHICH HAVE A FAITHFUL REPRESENTATION OF DEGREE LESS THAN .p 1=2/

Abstract: In a previous paper [5], the author and J. G. Thompson investigated groups which satisfy the more stringent condition p > 2« +1. The general outline of the proof of the theorem above is quite similar to the proof used in [5]. However, since there are two possibilities in the conclusion of the theorem, induction of necessity forces one to investigate several cases where only one occurred in the argument used in [5]. The theorem is of course vacuously true if « is a prime. However, in the course of the proof it is often necessary to use detailed information concerning the irreducible characters of a group whose order is divisible by only the first power of a prime. The relevant information which is needed is contained in §3 and is essentially due to R. Brauer [2; 3] and H. F. Tuan [7]. At this point the theory of modular characters appears to be unavoidable. We also utilize the fact proved by N. Ito [6], that if a solvable group G satisfies the hypotheses of the theorem, then G contains a normal subgroup of order h.

On $N$-high subgroups of Abelian groups.

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