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Showing papers in "Pacific Journal of Mathematics in 1977"




Journal ArticleDOI
TL;DR: In this paper, the authors generalize M. G. Kreϊn's formula for the generalized resolvents of a symmetric operator to the symmetric linear relation case.
Abstract: In some problems related to the spectral theory in Hubert space it is more natural and at the same time often less restrictive to use symmetric linear relations (in the terminology of [1], subspaces in the terminology of [2-4]) instead of symmetric operators. Hence the question arises if the theory of generalized resolvents of symmetric operators can be extended to symmetric linear relations. In [4] a description of all generalized resolvents of a symmetric linear relation was given, following the lines of A. V. Straus [5] in the operator case. It is the aim of this paper to generalize M. G. Kreϊn's formula for the generalized resolvents of a symmetric operator (see [6, 7]) to the symmetric linear relation case. This can be done rather easily by means of the Gayley transformation, using the results of [8]. However, in this connection there arise natural problems and questions: To introduce and to study the Q-function of a linear relation, to prove criteria for the selfadjoint extension of the given symmetric linear relation being an operator, to study the special case of a bounded nondensely defined operator etc. After the necessary definitions and their simple consequences in §1, the §2 is devoted to a study of the Q-f unction. From arguments similar to those in [9, 10] it follows that every function Q, whose values are bounded operators in a Hubert space and which is holomorphic in the upper half plane and has the property

189 citations







Journal ArticleDOI
TL;DR: In this paper, it was shown that the operators Δu and J can be associated with any fairly general real subspace of a complex Hubert space, and that many of their properties, for example the characterization of Au in terms of the K.M.S. condition, can be derived in this less complicated setting.
Abstract: of the paper we show that the operators Δu and J can, in fact, be associated with any fairly general real subspace of a complex Hubert space, and that many of their properties, for example the characterization of Au in terms of the K.M.S. condition, can be derived in this less complicated setting. In the second half of the paper we show, by using some of the ideas from the first half, that a simplified proof of the Tomita-Takesaki theory given recently by the second author can be reformulated entirely in terms of bounded operators, thus further simplifying it by, among other things, eliminating all considerations involving domains of unbounded operators.

89 citations








Journal ArticleDOI
TL;DR: In this paper, it was shown that the natural Hausdorff-young inequality on the "ax + 6" group has a constant less than 1 (Prop. 1) and that it behaves in some respects like the Fourier transform on Abelian groups.
Abstract: In recent work the author stated an inequality of Hausdorίϊ-Young type for integral operators which proved to be useful in obtaining Lp estimates on certain locally compact unimodular groups. The present paper is devoted to a closer analysis of that inequality together with some applications to operator theory and to Lp-Fourier analysis on locally compact groups. In the first place, the proof given previously in [15 I], for the inequality is incomplete, so this paper will begin in §2 with a correct proof of the inequality (Theorem 1). Also shown in §2 is the nonexistence of extremal functions in a particular instance (Prop. 8). In §3 the results of §2 are applied to obtain estimates for the norm of the Lp-Fourier transform on certain unimodular groups. Here some of the machinery from [10] is used in the examples, one class of which (Prop. 13) does not depend on Theorem 1. This has happened before, see [151: §3]. For certain members of this class however, it is shown that a better estimate can be obtained using Theorem 1 (Prop. 15). In §4 the study of Hausdorff-Young inequalities on nonunimodular groups is initiated. In view of the recent work on Plancherel formulas for nonunimodular groups such inequalities with constant 1 might be considered routine. It is shown here using Theorem 1, that the natural Hausdorfϊ-Young inequality on the "ax + 6" group has a constant less than 1 (Prop. 19). In §5 an operator valued analog of the Fourier transform on Abelian groups is introduced, which is motivated by preceding sections, and it is shown, using Theorem 1, that it behaves in some respects like the Fourier transform (Prop. 20).


Journal ArticleDOI
TL;DR: In this article, arithmetic properties of the function A(n) = 7,,,,,,ap have been discussed and congruences involving the partition of integers into primes have been considered.
Abstract: We discuss in this paper arithmetic properties of the function A(n) = 7,,,,,,ap . Asymptotic estimates of A(n) reveal the connection between A(n) and large prime factors of n . The distribution modulo 2 of A(n) turns out to be an interesting study and congruences involving A(n) are considered . Moreover the very intimate connection between A(n) and the partition of integers into primes provides a natural motivation for its study .



Journal ArticleDOI
TL;DR: In this paper, the basic theorem for group members in a ring was used to show locally that a ring element a e R (with unity) is unit regular exactly when there is a unit ue R and a group G in R such that a e uG is a group member.
Abstract: 1* Introduction* It is well-known that [15, 7] a ring R is strongly regular if and only if every aeR is a group member. In this note we shall use the basic theorem for group members in a ring to show locally that a ring element a e R (with unity) is unit regular exactly when there is a unit ue R and a group G in R such that a e uG. Hence unit regular rings are, as it were locally a "rotated" version of strongly regular rings. We remind the reader that a ring R is called regular if for every aeR, aeaRa; strongly regular if for every aeR, aeaR, and unit regular if for every aeR, there is a unit u e R such that ana = a [3]. Similar definitions hold locally. A ring with unity is called finite if ah = 1 implies ha — 1. Any solution a~ to axa = a is called an inner or 1-inverse of [1], while any solution a to axa = a and xax = x is called a reflexive or 1-2 inverse of a. For idempotents e and / in R, e ~ / denotes the equivalence in



Journal ArticleDOI
TL;DR: In this paper, a characterizati on self-adjointness for closed, strongly cyclic representations is presented, and a general class of representatio ns, called adjointable representations, is introduced and irreducibility of representations is considered.
Abstract: Basic results on unbounded operator algebras are given, a general class of representatio ns, called adjointable representations is introduced and irreducibility of representations is considered. A characterizati on of self-adjointness for closed, strongly cyclic ^-representations is presented.