Abstract: A class of Mikusiriski operators, called regular operators, is studied. The class of regular operators is strictly smaller than the class of all operators, and strictly larger than the class of all distributions with left bounded support. Regular operators have local properties. Lions' theorem of supports holds for regular operators with compact support. The fundamental solution to the Cauchy- Riemann equations is not regular, but the fundamental solution to the heat equation in two dimensions is regular and has support on a half-ray.

Abstract: Our basic goal is to develop an index theory for almost periodic pseudo-differential operators on R ~. The prototype of this theory is [5] which has direct application to the almost periodic Toeplitz operators. Here, we s tudy index theory for a C*-algebra of operators on R ~ which contains most almost periodic pseudo-differential operators such as those arising in the s tudy of elliptic boundary value problems for constant coefficient elliptic operators on a half space with almost periodic boundary conditions. Our program is as follows: We begin with a discussion of a C*-algebra with symbol which contains all of the classical pseudo-differential operators on R ~. Precisely, if A is a bounded operator on L2(R ~) and 2 ER ~, let e~(A) denote the conjugate of A with the function e *~'x acting as a multiplier denoted e~. We first s tudy the C*-algebra of those A for which the function 2~+e~(A) has a strongly continuous extension to the radial compactification of R ~. The restriction of this function to the complement of R ~ then gives the usual (principal) symbol a(A) when A is a pseudo-differential operator of order zero (of a suitable type). We characterize the Fourier multipliers in this algebra and the image of the symbol map. We give sufficient conditions for the usual construction of a pseudo-differential operator as well as one of Friedrichs' constructions to give an element of this algebra. In particular, the lat ter gives a positive linear right inverse for the symbol m a p a t least when the symbol is sufficiently smooth. I n fact, we show in w 3 tha t the Friedrichs map is a right inverse to the symbol map in the almost periodic case. We expect this to be true in the general case also.

Abstract: In this paper it is shown that if D and E are con- tinuous linear operators on a Banach space X, then the following are equivalent: (i) B is a right factor of A, (ii) B majorizes A and (iii) the range of B* contains the range of A*.

Abstract: A decomposition of some operators is obtained that represents a generalization of Brown's (1953) work on quasi-normal operators. It is shown that the decomposition obtained is of most interest when the operator considered is far from being finite-dimensional. An application of the results obtained to the study of quasi-triangular operators is presented for illustration.

Abstract: A graphical representation of matrix elements of spin-free one- and two-electron operators is used for deriving a simple algorithm for the evaluation of their values. The method covers all the cases which may occur when wave functions are taken as mutually orthogonal antisymmetrized products of spinorbitals (which are assumed to form an orthonormal set) and are eigenfunctions of L 2 and L z operators. The resulting formulas are suitable as well for computer programming as for hand calculations.

Abstract: Call an operator A with domain and range in a complex Banach space X hermitian if and only if iA generates a strongly continuous one-parameter group of isometries on X. Hermitian operators in the Hardy spaces of the disc (HP, 1 < p < oo) are investigated, and the following results, in particular, are obtained. For 1 < p < oo, p '0 2, the bounded hermitian operators on HP are precisely the trivial ones-i.e., the real scalar multiples of the identity operator. Furthermore, as pointed out to the authors by L. A. Rubel, there are no unbounded hermitian operators in H'. To each unbounded hermitian operator in the space HP, 1 < p < oo, p # 2, there corresponds a uniquely determined one-parameter group of conformal maps of the open unit disc onto itself. Such unbounded operators are classified into three mutually exclusive types, an operators type depending on the nature of the common fixed points of the associated group of conformal maps. The hermitian operators falling into the classification termed "type (i)" have compact resolvent function and one-dimensional eigenmanifolds which collectively span a dense linear manifold in HP.

Abstract: With a view to obtaining an orthogonal solution to the state labelling problem of SU(3) in an O(3) basis, four independent operators which shift the eigenvalues of the O(3) Casimir operator L2 are constructed. The hermiticity properties of these operators, and of certain of their products which commute with L2, are discussed.

Abstract: a) A characterization of extreme operators (in the unit ball of operators) between L 1'-spaces is given, together with other related proper- ties. (b) A general theorem of Kreln-Milman type for the unit ball of operator spaces is proved, and is applied to operators between Ll-spaces and to oper- ators into C-spaces. 1. Introduction. In this paper we study in detail extreme points in the unit ball of operator spaces (which we shall briefly call extreme operators). Although we treat mostly operators in L '-spaces and C-spaces, some of the results are general (?3). Before discussing the results, we go briefly over the notations. They are mostly standard and we refer to (14) for some of the notations, though we remark that if E is a Banach space, the action of x* E E* on x E E will be denoted either by x*(x) or by (x*, x). The canonical imbedding of E into E** will be denoted by JE. The scalars are either real or complex; only in Theorem 2.10 are they assumed to be real. Generalizing a notation of Morris and Phelps (121, we call an operator T: E -. F, for two Banach spaces E and F, a nice operator, if T*(ext S(F*)) C ext S(E*). It is easy to verify that every nice oper- ator is extreme, and to construct examples where there exist no nice operators

Abstract: Introduction. Following Solovay [2], let 'ZF' denote the axiomatic set theory of Zermelo-Fraenkel and let 'ZF + DC' denote the system obtained by adjoining a weakened form of the axiom of choice, DC, (see p. 52 of [2] for a formal statement of DC). From DC a 'countable' form of the axiom of choice is obtainable. More precisely, if {Bn:n e N} is a countable collection of nonempty sets then it follows from DC that there exists a function ƒ with domain N such that f(n) e Bn for each n. The system ZF -h DC is important because all the positive results of elementary measure theory and most of the basic results of elementary functional analysis, except for the Hahn-Banach theorem and other such consequences of the axiom of choice, are provable in ZF + DC. In particular, the Baire category theorem for complete metric spaces and the closed graph theorem for operators between Fréchet spaces are provable in ZF + DC. Solovay shows [2] that the proposition, Each subset of the real numbers is Lebesgue measurable, cannot be disproved in ZF + DC. He does this by constructing a model for ZF + DC in which the proposition becomes a true statement. We shall see that the proposition, Each linear operator on a Hilbert space is a bounded linear operator, is consistent with the axioms of ZF + DC. Other results of this type are obtained. For example, Whenever X and Y are separable Fréchet groups and h:X -• Y is a homomorphism then h is continuous, cannot be proved or disproved in ZF + DC. Fortunately all the hard work in model theory has been done by Solovay. All that we use here is straightforward functional analysis.

Abstract: ()• Introduction* One of the objects of the present paper is to study the eigenvalue problem Tx = xAx by means of a two (linear) operator or \"bioperative\" numerical range. We recall that Toeplitz [30] defined the numerical range for matrices in 1918. Then Wintner [31] in 1930 and Stone [27], [28] in 1930 and 1932 discussed the relationship between the convex hull of the spectrum of a bounded linear operator on a Hubert space and its numerical range. In 1943, J. Dieudonne [4] and S. M. NikoPskii [24] laid the groundwork which would later help to show that the index of a bounded semi-Fredholm operator is stable under perturbation by a bounded linear operator of sufficiently small norm. This result was established in 1951 for bounded operators on a Hubert space by F.V.Atkinson [2] and (independently) I. C. Gohberg [9], [10], and [11]. The following year, M. G. Krein and M. A. KrasnosePskii [20], B. Sz.-Nagy [29], and I. C. Gohberg [12] generalized these results to unbounded closed linear operators. M. G. Krein and M. A. KrasnosePskii also established the semi-stability of the nullity and deficiency. To understand the foundations and historical development of the whole theory, the reader is referred to the comprehensive article of Gohberg and Krein [13] which appeared in 1957. Among the many innovations appearing in the 1958 paper of T. Kato [18] was the concept of the \"lower bound\" (now called the \"minimum modulus\") of a linear operator A defined on a Banach space. The main reason for defining the minimum modulus of A was to obtain as small a disc about the origin as possible so that ind(Ύ— XA) = ind(A) for all λ outside that disc, A being semi-Fredholm and T being bounded or relatively bounded with respect to A. In this paper we introduce a bioperative numerical range which will improve that result for Fredholm operators defined on a Hubert space. The improvement is a consequence of the fact that the bioperative numerical range for Fredholm A and relatively bounded T defined on a Hubert space is always contained in the aforementioned disc and that the index of T — XA remains constant if λ is not in the closure of that bioperative numerical range. For another use of the bioperative numerical range, we recall

Abstract: The linear algebra and combinatorial aspects of the Rota-Mullin theory of polynomials of binomial type are separated and the former is developed in terms of shift operators on infinite dimensional vector spaces with a view towards application in the calculus of finite differences.