Louis de Branges

The Stone-Weierstrass theorem

Let S be a locally compact Hausdorff space and let C(S) be the continuous complex valued functions which (in the noncompact case) vanish at infinity. Let E be a vector subspace of C(S) which is closed under complex conjugation. We ask for conditions that E be uniformly dense in C(S). (The hypothesis that E be closed under complex conjugation allows us to reduce the problem to one for real valued functions. The reader may prefer to recast the discussion in that context.) We have chosen to present our result as a lemma towards a proof of the Stone-Weierstrass theorem. This approach not only sheds an interesting light on that theorem, but will help the reader understand the nature of the lemma. However, the discussion of Stone [3] remains the most direct and, when all details are considered, the shortest proof of the Stone-Weierstrass approximation theorem. A more serious application of the lemma will be made later in a paper on the Bernstein approximation problem. Let U(E) be the set of all real valued measures A on the Borel subsets of S, with total variation at most 1, such that for every f in E, ffdlu = 0. Consider U(E) in the weak topology induced by C(S) under integration. Then U(E) is a compact, convex set, and if E is not dense in C(S), U(E) contains a nonzero element (Loomis [2, pp. 22-23, pp. 29-47]). By the Krein-Milman theorem, U(E) is the weakly closed convex span of its extreme points (Krein-Milman [11).

SOME HILBERT SPACES OF ENTIRE FUNCTIONS. IV

Recent work has been concerned with Hilbert spaces whose elements are entire functions and which have these three properties: (HI) Whenever F(z) is in the space and has a nonreal zero w, the function F(z) (z 0)/(z w) is in the space and has the same norm as F(z). (H2) Whenever w is a nonreal number, the linear functional defined on the space by F(z) -, F(w) is continuous. (H3) Whenever F(z) is in the space, the function F*(z) = F(z) is in the space and has the same norm as F(z). If E(z) is an entire function which satisfies the inequality

The Bernstein problem

Let w(x) be a positive valued, continuous function of real x, such that for each n=O, 1, 2, , xnW(X) is bounded. S. Bernstein asks for conditions on w(x) that the weighted polynomials P(x)w(x) be uniformly dense in the continuous complex valued functions which vanish at infinity (Pollard [5]). THEOREM. A necessary and sufficient condition that the weighted polynomials P(x)w(x) fail to be dense in the continuous functions which vanish at infinity is that there be an entire function F(z) of exponential type, not a polynomial, which is real for real z and whose zeros XA are real and simple, such that log+ I F(t) .dt < oo 1 + t2 and

Complementation in Krein Spaces

A generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product. Complementation is a construction in the geometry of Hilbert space which was applied to the invariant subspace theory of contractive transformations in Hilbert space by James Rovnyak and the author [6]. The concept was later formalized by the author [3]. Continuous and contractive transformations in Krein spaces appear in the estimation theory of Riemann mapping functions [4]. It is therefore of interest to know whether a generalization of complementation theory applies in Krein spaces. Such a generalization is now obtained. The results are also of interest in the invariant subspace theory of continuous and contractive transformations in Krein spaces [5]. The vector spaces considered are taken over the complex numbers. A scalar product for a vector space )1 is a complex-valued function (a, b) of a and b in )1 which is linear, symmetric, and nondegenerate. Linearity means that the identity (aa + 3b, c) = a(a, c) + 3(b, c) holds for all elements a, b, and c of )1 when a and , are complex numbers. Symmetry means that the identity (b, a) = (a, b) holds for all elements a and b of )1. Nondegeneracy means that an element a of )1 is zero if the scalar product (a, b) is zero for every element b of )1. Every element b of )1 determines a linear functional bon )1 which is defined by b-a = (a, b) for every element a of )1. The weak topology of )1 is the weakest topology with respect to which bis a continuous linear functional on )1 for every element b of )1. The weak topology of )1 is a locally convex topology having the property that every continuous linear functional on )1 is of the form bfor an element b of )1. The element b is then unique. The antispace of a vector space with scalar product is the same vector space considered with the negative of the given scalar product. A fundamental example of a vector space with scalar product is a Hilbert space. A Krein space is a vector space with scalar product which is the orthogonal sum of a Hilbert space and the antispace of a Hilbert space. Received by the editors October 10, 1986 and, in revised form, February 23, 1987. The results of the paper were presented to the Department of Mathematics, Indiana and Purdue University in Indianapolis, on March 27, 1987, as the Ernest J. Johnson Colloquium. 1980 Mathematics Subject Classification (1985 Revision). Primary 46D05. Research supported by the National Science Foundation. ?1988 American Mathematical Society 0002-9947/88

Some Hilbert spaces of analytic functions. III

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The Riemann hypothesis for Hilbert spaces of entire functions

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Krein spaces of analytic functions

This paper continues the study of Hilbert spaces of analytic functions which are involved in the structural analysis of nonself-adjoint transformations in Hilbert space. The theory of nonself-adjoint transformations originates in the quantum mechanical problems of nuclear scattering theory. Although the ordinary use of the Schriidinger equation leads to self-adjoint transformations, it is customary for physicists to subdivide the nuclear reaction in such a way that nonself-adjoint transformations occur. This observation, made by Livgic [l], caused him to found a general theory of nonself-adjoint transformations. Although LivBic’s theory is extensive and powerful, it has received very little recognition because no one, apparently, can follow his arguments. The difficulties are so great that Dolph and Penzlin [2] have attempted an independent derivation of the main results. The trouble is due not so much to logical gaps as it is to insufficient motivation of the main tool, the characteristic operator function. This quantity arises in the description of fundamental solutions of formally self-adjoint differential equations under variations of the boundary conditions. Yet the characteristic operator function is applied to transformations which have no connection with differential equations. The trouble is that LivGc has missed the meaning of the characteristic operator function, which is to be found in the construction of certain Hilbert spaces of analytic functions [3]. To explain how these spaces originate, we must go back to the basic work of Stone [4] and to our previous work with entire functions [5-81. Stone’s book has two different objectives, apart from a general formulation of concepts. The first is the study of self-adjoint transformations. It is important to note how he goes about the study of these transformations. From his point of view the structure theorem is an abstract analogue of the integral representation of functions which are analytic and have a nonnegative real part in the upper half-plane. (See, for example, Nevanlinna and Nieminen [9] for a more detailed reconstruction of the same argument.) It has

Tensor product spaces

History of the scattering method. The scattering theory of linear differential operators is an approach to their spectral analysis which arises from the study of wave propagation. The medium in which waves are propagated is regarded as a linear system whose input comes from the remote past and whose output is given in the distant future. The scattering operator describes the energy-preserving transition from past to future. The awareness of a relationship between scattering theory and that area of number theory associated with the Riemann hypothesis is at least twenty-five years old. It is found for example in an analysis of the Laplace-Beltrami operator by Ehrenpreis and Mautner [4] and in a congress address of Gelfand [6]. In the same years the author constructed the Hubert spaces of entire functions which are used in the present formulation of scattering theory. In 1972, Faddeev and Pavlov [5] applied the Lax-Phillips scattering theory to the Laplace-Beltrami operator, considered in a space of functions invariant under the action of the modular group. An account of their result in English, and a generalization, appeared in 1976 in a monograph by Lax and Phillips [7]. The important observation is made that the Riemann hypothesis is equivalent to decay properties in the wave propagation associated with the operator. But no geometric reason could be found for the propagation to have these

The convergence of Euler products

Abstract An invariant subspace theory for contractive transformations in Hilbert spaces which is due to L. de Branges and J. Rovnyak (in “Perturbation Theory and Its Applications in Quantum Mechanics,” pp. 295–391, Wiley, New York, 1966 ) is generalized. Observable, contractive, and conjugate-isometric linear systems are considered whose state spaces and coefficient spaces are Krein spaces. Such linear systems have canonical models with state spaces chosen as Krein spaces whose elements are vector-valued analytic functions. The properties of these spaces are related to the theory of square summable power series with coefficients in a Krein space. Generalizations of a celebrated theorem of A. Beurling (Acta Math.81 (1949) , 209–255) and P. D. Lax (Acta Math.101 (1959) , 163–178) are obtained in that context. An underlying theme is the relation between factorization and invariant subspaces for continuous and contractive transformations in Krein spaces. Known results for Hilbert spaces ( L. de Branges, J. Math. Anal. Appl.29 (1970) , 168–200) are generalized. Some of these results have previously been generalized to Pontryagin spaces by D. Alpay and H. Dym (in “Operator Theory: Advances and Applications,” Vol. 18, pp. 89–159, Birkhauser-Verlag, Basel, 1986 ).

Some mean squares of entire functions

A decomposition is obtained for the tensor product of two irreducible representations of the group of conformal mappings of the upper half-plane when one representation is taken in a Hilbert space of analytic functions and the other in the conjugate of such a space. An estimation theory for modular forms results which is of interest in connection with the Riemann hypothesis [l]. The decomposition depends on an eigenfunction expansion associated with Gauss’s hypergeometric function [2]. Let Y be a given number, Y > -1. The Hardy space 9” is the set of functions F(z), analytic in the upper half-plane, of the form