Paul R. Halmos

The Marriage Problem

In a recent issue of this journal Weyl1 proved a combinatorial lemma which was apparently considered first by P. Hall2 Subsequently Everett and Whaples 3 published another proof and a generalization of the same lemma. Their proof of the generalization appears to duplicate the usual proof of Tychonoff’s theorem.4 The purpose of this note is to simplify the presentation by employing the statement rather than the proof of that result. At the same time we present a somewhat simpler proof of the original Hall lemma.

How to teach

Even the most junior departmental secretaries scared me when I first arrived in Chicago—they represented officialdom. If I asked one to do a five-minute duplicating job, and she said she’d have it for me the day after tomorrow, I said yes, sure, all right. I didn’t really know how long the job should take, and I was brought up never to argue with the government. By the time I left Chicago I had learned that secretaries were human and that they were not always right ex officio.

Products of involutions

Abstract Every square matrix over a field, with determinant ±1, is the product of not more than four involutions.

The Foundations of Probability

(1944). The Foundations of Probability. The American Mathematical Monthly: Vol. 51, No. 9, pp. 493-510.

Commutators of operators

A mathematical formulation of the famous Heisenberg uncertainty principle is that a certain pair of linear transformations P and Q satisfies, after suitable normalizations, the equation PQ - QP = 1. It is easy enough to produce a concrete example of this behavior; consider L2(-∞, +∞) and let P and Q be the differentiation transformation and the position transformation, respectively (that is, (Pf) (x) = f’(x) and (Qf) (x) = xf(x)). These are not bounded linear transformations, of course, their domains are far from being the whole space, and they misbehave in many other ways. Can this misbehavior be avoided?

Some unsolved problems of unknown depth about operators on Hilbert space

The paper presents a list of unsolved problems about operators on Hilbert space, accompanied by just enough definitions and general discussion to set the problems in a reasonable context. The subjects are: quasitriangular matrices, the resemblances between normal and Toeplitz operators, dilation theory, the algebra of shifts, some special invariant subspaces, the category (in the sense of Baire) of the set of non-cyclicoperators, non-commutative(i.e. operator) approximation theory, infinitary operators, and the possibility of attacking invariance problems by compactness or convexity arguments.

The Basic Concepts of Algebraic Logic

(1956). The Basic Concepts of Algebraic Logic. The American Mathematical Monthly: Vol. 63, No. 6, pp. 363-387.

Recent progress in ergodic theory

Prologue. In 1948, at the November meeting of the Society in Chicago, I delivered an address entitled Measurable transformations. In the twelve years that have elapsed since then, ergodic theory (of which the theory of measurable transformations is the greatest part) has been spectacularly active. The purpose of todays address is to report some of the developments of those twelve years ; its title might well have been Measurable transformations revisited. The subjects I chose for this purpose are: some new ergodic theorems, information theory and its connection with ergodic theory, and the problem of invariant measure. The stage on which most ergodic performances take place is a measure space consisting of a set X and of a measure JJL denned on a specified cr-field of measurable subsets of X. At the most trivial level X consists of a finite number of points, every subset of X is measurable, and jit is a mass distribution on X (which may or may not be uniform). At a more useful and typical level X is the real line ( c o , + <*>), or the unit interval [0, l ] , measurability in either case is interpreted in the sense of Borel, and ju is Lebesgue measure. Another possibility is to consider a measure space having a finite number of points with total measure 1 and to let X be the Cartesian product of a countably infinite number of copies of that space with itself; measurability and measure in this case are interpreted in the customary sense appropriate to product spaces. This latter example is easily seen to be measuretheoretically isomorphic to the unit interval, as also are most of the normalized measure spaces (measure spaces with total measure 1) that ever occur in honest analysis. The only measure spaces I shall consider in this report are the ones isomorphic to one of the spaces already mentioned. The expert will know just how little generality is lost thereby, and the casual passer-by, quite properly, will not care. A transformation T from a measure space X into a measure space Y is called measurable if the inverse image T~E (in X) of each measurable set E (in F) is again a measurable set. A measurable transformation T is measure-preserving if, for every measurable set £ , the sets E and T~E have the same measure. A measurable (but not

Ten years in Hilbert space

This is a report on progress in the theory of single operators in the 1970s. It is based for the most part, but not exclusively, on ten problems in Hilbert space posed in 1970 [21]; it reports which of those problems have been solved and what the solutions are. It reports some closely related results also, notably those of Apostol, Foiaş, and Voiculescu on the spectral characterization of non-quasitriangular operators, Scott Brown on invariant subspaces of subnormal operators, Gambler on invariant subspaces of some Toeplitz operators, Kriete and Trutt on the subnormality of the Cesàro operator, and Lomonosov on hyperinvariant subspaces of compact operators.

Algebraic logic. IV. Equality in polyadic algebras

Introduction. A standard way to begin the study of symbolic logic is to describe one after another the propositional calculus, the monadic functional calculus, the pure first-order functional calculus, and the functional calculus with equality. The algebraic aspects of these logical calculi belong to the theories of Boolean algebras, monadic algebras, polyadic algebras, and cylindric algebras respectively. The connection between the propositional calculus and Boolean algebras is well-known; for a recent exposition of it (and also of some aspects of the more advanced theories) see The basic concepts of algebraic logic, Amer. Math. Monthly vol. 63 (1956) pp. 363-387. Monadic algebras and polyadic algebras were studied in the first three papers of this sequence; [see Algebraic logic III, Trans. Amer. Math. Soc. vol. 83 (1956) pp. 430-470, and the references given there]. Cylindric algebras were introduced by Tarski and Thompson [Some general properties of cylindric algebras, Bull. Amer. Math. Soc. Abstract 58-1-85; see also Tarski, A representation theorem for cylindric algebras, Bull. Amer. Math. Soc. Abstract 58-1-86.] Most of what was done for polyadic algebras in [ll] and [ill] was restricted to locally finite polyadic algebras of infinite degree. (The Roman numerals refer to the other parts of this sequence.) Since it is known that every locally finite cylindric algebra of infinite degree possesses a natural polyadic structure, the results of those papers apply to cylindric algebras without any change. This paper, on the other hand, is mostly pre-cylindric; its main purpose is to discuss (in algebraic language) the introduction of equality. The paper is not self-contained. The notation introduced in §1 of [ill] will be used without any further explicit reference, and some of the basic concepts studied in [ill] (notably the concept of a predicate) will also be assumed known. (The most difficult part of [ill], the theory of terms, is used in §9 only.) At one point (§6) the representation theorem for simple polyadic algebras is needed, and later (§7) we make use of the duality theory for monadic algebras. Most parts of the paper, however (and, in particular, all definitions and the statements of all the theorems) are accessible to anyone who has skimmed through [III], provided that, in addition, he is acquainted with the elementary theory of functional polyadic algebras. For the con-