1.1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. Suppose that ∑ an (we write ∑ for throughout the present paper) is a given infinite series and let

The series ∑ an is said to be summable |R, λ, r|, where r > 0, if

where A is a fixed positive number (6, Definition B). Now, for r > 0, m < ω < m + 1,

Hence, ∑ an is summable |R, λ, r|, where r > 0, if

1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonie increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series an is summable |R, λ, r|, where r > 0, if

where A is a fixed positive number (6, Definition B).

Let f(t) be a periodic function with period 2π and Lebesgue integrable over (–π, π) and let

1.1

The series conjugate to (1.1), at t = x, is

1.2

Let D be an integrally closed domain with identity having quotient field L. If {Vα } is the set of valuation overrings of D and if A is an ideal of D, then à = ∪ α AVα is an ideal of D called the completion of A. If X is an indeterminate over D and f ∈ D[X], then we denote by Af the ideal of D generated by the coefficients of f. The Kronecker function ring DK of D is defined by DK = {f/g| f, g ∈ D[X], Ãf ⊆ Ag } (4, p. 558); and the domain D(X) is defined by D(X) = {f/g| f, g ∈ D[X], Ag = D} (5, p. 17). In this paper we wish to relate the ideal theory of D to that of DK and D(X) for the case in which D is a Prüfer domain, a Dedekind domain, or an almost Dedekind domain.

Let (X, T, π) denote a flow, where X is a compact topological space metrizable by d, and T is a closed non-trivial subgroup of the reals under addition. T is recurrent if and only if for each and s > 0, there exists t > s such that x ∈ X implies . If T is almost-periodic, then T is both recurrent and distal. In §§ 4 and 5, it is shown that, under more stringent hypotheses, the recurrence of T is neither a necessary nor a sufficient condition for T to be distal. Let S be a closed non-trivial subgroup of T. It is shown in § 3 that T is recurrent if and only if S is recurrent. From this result, we obtain a solution to a problem posed by Nemyckiĭ (16, p. 492, Problem 6).

Associated with any integral domain R there is a partially ordered group A, called the group of divisibility of R. When R is a valuation ring, A is merely the value group; and in this case, ideal-theoretic properties of R are easily derived from corresponding properties of A, and conversely. Even in the general case, though, it has proved useful on occasion to phrase a ring-theoretic problem in terms of the ordered group A, first solve the problem there, and then pull back the solution if possible to R. Lorenzen (15) originally applied this technique to solve a problem of Krull, and Nakayama (16) used it to produce a counterexample to another question of Krull. More recently, Heinzer (7;8) has used the method to construct other interesting examples of rings.

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).

Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.

2. THEOREM. Let X be a complex Banach space. If T belongs to the class , and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

It is well known that the distribution of the zeros of an analytic function affects its rate of growth. The literature is too extensive to indicate here. We only point out (1, p. 27; 2; 3; 5), where the angular distribution of the zeros plays a role, as it will in this paper. In private communication, A. Zygmund has raised the following related question, which is the subject of our investigation here.

Let {zn }, n = 1, 2, 3, …, be a sequence of non-zero complex numbers of modulus less than 1, such that ∑(1 – |zn |) < ∞, and consider the Blaschke product

1

Let

2

What are the sequences {zn } for which I(r) is a bounded function of r?

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called a sequence space. A sequence space E is a BK-space (9) if it is equipped with a norm under which: first, E is a Banach space and second, each of the coordinate maps x → xi is continuous. Let ∑ be the group of all permutations of Z + = {1, 2, 3, …}. If x ∈ ω and σ ∈ ∑, the sequence x σ is defined by (x σ)i = x σ(i)). A sequence space E is symmetric if x σ ∈ E whenever x ∈ E and σ ∈ ∑. Accounts of symmetric sequence spaces occur in (3; 7; 8).

Let R be a field of rational functions of one variable over a field of constants R 0. Dock Sang Rim (6) has proved that the global reciprocity law in exactly the usual sense holds whenever R 0 is an absolutely algebraic quasi-fini te field of characteristic not equal to 0: this was known before only when R 0 was a finite field. We shall give another proof of Rim's result by means of a noteworthy generalization of the usual global reciprocity law. Namely, let R 0 be a finite field and let F be the set of all fields k contained in some fixed R alg.clos. and of finite degree over R. The reciprocity law states that there exists a family {fk }, k ∈ F, of functions fk : Ck → G(k abel.clos./k) (where Ck is the idèle class group of k) enjoying certain properties such as the norm transfer law.

The investigation of finite semigroups S with quasi-Frobenius (q.-F.) algebras F(S) over a field F was begun in (7; 8). The problem for commutative semigroups was reduced (7, Theorem 3) to the study of semigroups of the form S = G ∪ S1, where G is a group and S1 is either the null set or is a nilpotent ideal in S (i.e., S1n = {0} for some positive integer n). Such semigroups were called “of type C”. The question is “When does a semigroup of type C have a q.-F. algebra over a field?” (7, Theorem 4) shows that no distinction need be made between the properties q.-F. and Frobenius for commutative algebras.

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp -spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp -space. The result is interesting for lp -spaces because of the paucity of standard bases in these spaces.

Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp -spaces, where is the space lt and the space ls with 1 < r <2 and 2 < s = r/(r – 1).

Let A1 and A2 be commutative Banach algebras and A1 ⊙ A2 their algebraic tensor product over the complex numbers C.There is always a t least one norm, namely the greatest cross-norm γ (2), on A1 ⊙ A2 that renders it a normed algebra. We shall write A 1 ⊗αA2 for the α-completion of A1 ⊙ A2 when αis an algebra norm on A1 ⊙ A2.Gelbaum (2; 3), Tomiyama (9), and Gil de Lamadrid (4) have shown that for certain algebra norms α on A1 ⊙ A2 every complex homomorphism on A1 ⊙ A2 is α-continuous. In § 3 of this paper, we present a condition on an algebra norm α which is equivalent to the α-continuity of every complex homomorphism on A1 ⊙ A2.

Let Lp,1 ≦ p< ∞, denote the space of all functions ƒ (real or complex) such that ƒ and |ƒ|p are variationally integrable (see 2, p. 40, for definition) with respect to a pair h of interval functions in an elementary set E. In what follows, we fix both h and E and assume that h = {hl ht}is variationally integrable and hs ≧ 0 (s = l, r)in E. Further, let L∞ denote the space of all functions ƒ (real or complex) which are h-measurable (cf. 2, p. 95) and bounded almost everywhere, i.e. except in a set X satisfying (cf. 2, p. 47) V(h; E; X)= 0.

If G = (G, τ) is a topological group with topology τ, then there is a smallest topology τ* ⊇ τ such that G* = (G, τ*) is a topological group with equal left and right uniformities (1). Bagley and Wu introduced this topology in (1), and studied the relationship between Gand G*. In this paper we prove some additional results concerning G* and groups with equal uniformities in general. The structure of locally compact groups with equal uniformities has been studied extensively. If G is a locally compact connected group, then G has equal uniformities if and only if G ≅ V× K,where F is a vector group and Kis a compact group (5). More generally, every locally compact group with equal uniformities has an open normal subgroup of the form V× K(4).

Let M n and Np be separable manifolds of dimensions n and p, respectively, with n ≧ p, and without boundary unless otherwise indicated. A mapƒ: M → N is proper if, for each compact set K ⊂ N, f –l(K) is compact. It is topologically equivalent to g: X → Y if there exist homeomorphisms α of M onto X and β of N onto Y such that βƒα–1 = g. At x ∈ M, ƒ is locally topologically equivalent to g if, for every neighbourhood W ⊂ M of x, there exist neighbourhoods U ⊂ W of x andV of ƒ(x) such that ƒ | U: U → V is topologically equivalent to g.

In this paper we give an extension of (6, Theorem 1), using a similar method of proof, to every homogeneous Siegel domain of second kind which can be mapped biholomorphically into a Kâhler manifold of a certain class (Theorem 1). Then by a well-known result of Vinberg, Gindikin, and Pjateckiï-Sapiro (10) that every bounded homogeneous domain D,contained in a complex euclidean space CN,can be mapped biholomorphically onto an affinely homogeneous Siegel domain of second kind, the theorem follows for D(Theorem 2). (6, Theorem 1) is a generalization of the Ahlfors version of the Schwarz-Pick lemma in C1 (1) to invariant volume for a star-like homogeneous bounded domain in CN;see also (4). In § 3 we give the inequality for a special non-symmetric Siegel domain of second kind using an explicit form of TD(z, )due to Lu (7).

In (6)Scholz asked if the inequality

1.1

held for all positive integers q, where l(n)is the number of multiplications required to raise xto the nth power (a precise definition of l(n)in terms of addition chains is given in § 2). Soon afterwards, Brauer (2) showed, among other things, that l(n) ∼(log n)/(log2). This suggests the problem of Calculating

1.2

It can be deduced from (2) that θ≦ 1. If θ <1, (1.1) follows immediately for infinitely many q.My main result,Theorem 5 of § 4, merely shows that θ is slightly larger than ⅓.Actually, I know of no case where (1.1) is not in fact an equality; a tedious calculation verifies this for 1 ≦ q≦ 8.

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that

(i) πi(Y) ∈ℭfor all i < n,

(ii) H*(X; Z) is finitely generated,

(iii) Hi(X;Z) ∈ℭfor all i > k.

Then the suspension homomorphism

is a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd ∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2

The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

If ℭ and ℭ′ are classes of finite abelian groups, we write ℭ + ℭ′ for the smallest class containing the groups of ℭ and of ℭ′. For any positive number r, ℭ < r is the smallest class of abelian groups which contains the groups Zp for all primes p less than r.

Our aim in this paper is to prove the following theorem.

THEOREM. Iƒ ℭ is a class of finite abelian groups and

(i) πi (Y) ∈ℭ for i < n,

(ii) H*(X; Z) is finitely generated,

(iii) Hi(X;Z)∈ ℭ for i > n + k,

Then

This statement contains many of the classical results of homotopy theory: the Hurewicz and Hopf theorems, Serre's (mod ℭ) version of these theorems, and Eilenberg's classification theorem. In fact, these are all contained in the case k = 0.

For a prime number p let be the class of finite abelian groups whose orders are prime to p. For a finitely generated abelian group G, let Gp be the sum of the free and p-primary components of G. Our aim in this paper is to prove the following theorem.

Theorem. Suppose that

(i) Hi(X;Z) = 0 for i > k,

(ii) for i > k – d

Then there exists a spectral sequence with

and the differential is given by