Let Σ be a σ-algebra of subsets of some set Ω and let μ:Σ→[0,∞] be a σ-additive measure. If Σ(μ) denotes the set of all elements of Σ with finite μ-measure (where sets equal μ-a.e. are identified in the usual way), then a metric d can be defined in Σ(μ) by the formula

here E ΔF = (E\F) ∪ (F\E) denotes the symmetric difference of E and F. The measure μ is called separable whenever the metric space (Σ(μ), d) is separable. It is a classical result that μ is separable if and only if the Banach space L1(μ), is separable [8, p.137]. To exhibit non-separable measures is not a problem; see [8, p. 70], for example. If Σ happens to be the σ-algebra of μ-measurable sets constructed (via outer-measure μ*) by extending μ defined originally on merely a semi-ring of sets Γ ⊆ Σ, then it is also classical that the countability of Γ guarantees the separability of μ and hence, also of L1(μ), [8, p. 69].

All polynomials considered in this paper belong to ℚ[x] and reducibility means reducibility over ℚ. It has been established by one of us [5] that every binomial in ℚ[x] has an irreducible factor which is either a binomial or a trinomial. He has further raised the question “Does there exist an absolute constant K such that every trinomial in ℚ[x] has a factor irreducible over ℚ which has at most K terms (i.e. K non-zero coefficients)?”

In [4], Ramanujan stated the following beautiful identity which was later proved by Bailey [2] and [3];

We denote by S(n) the coefficient of qn on the right hand side of (1). The object of this paper is to prove the following two theorems.

We identify the extreme points of the unit sphere of the Lorentz space Lw,1 This yields a characterization of the surjective isometries of Lw,1(0,1). Our main result is that every element in the unit sphere of Lw,1 is the barycenter of a unique Borel probability measure supported on the extreme points of the unit sphere of Lw,1.

Let I denote the interval [0,1] in its usual topology and let S = S(I) be the semigroup of continuous mappings of I into itself with function composition as the semigroup operation. In a survey talk given at Oberwolfach in 1989 (see [5]), K. D. Magill Jr. pointed out that some elementary algebraic properties of S are still unknown. We shall answer one of the questions he raised (which appears as Problem 4.6 in [5] and which he had asked earlier, as long ago as 1975 [3]) by showing that S has infinitely many distinct two-sided ideals. In fact, we shall produce an infinite descending sequence of distinct ideals. As Magill points out, this also solves his Problem 4.5: S has infinitely many distinct congruences. We believe that S must have c distinct ideals, but we have been unable to prove this.

We prove that sufficiently separated sequences are interpolating sequences for f′(z)(1−|z|2) where f is a Bloch function. If the sequence {zn} is an η net then the boundedness f′(z)(1−|z|2) on {zn} is a sufficient condition for f to be a Bloch function. The essential norm of a Hankel operator with a conjugate analytic symbol acting on the Bergman space is shown to be equivalent to .

Let K be a field of characteristic different from 2, and a, b quadratically independent elements of K. Put J= K(√a, √b). In [4], Jensen and Yui discuss the question of quaternionic (Q8) extensions of J, and give a survey of known results. In [8], Ware discusses (among other things) some general conditions for, and relations between, the existence of Q8 and D4 (dihedral) extensions of K. A general theorem of Witt [9] says that J will have a quaternionic extension J(√u) if and only if there exists a 3 × 3 matrix P over K such that PPt = diag(a, b, 1/ab), and an appropriate value for u is given in terms of the entries of P. The problem of actually finding P in a particular case is not trivial.

Many of the classical inequalities of analysis can be written in the form P(x) ≥ 0 for x ∈ I or P(x) > 0 for x ∈ I′, where P(x) is a polynomial and I′ ⊂ I are certain intervals on the real line. This gives rise to the question of where the zeros of P(x) are located. For example, if f is a polynomial with real zeros, then an inequality of Laguerre [8, p. 171 f.] asserts that

for all x. A detailed study of the zeros of this particular P(x) has been made [5].

The concept of weak relative-injectivity of modules was introduced originally in [10], where it is shown that a semiperfect ring R is such that every cyclic right module is embeddable essentially in a projective right R-module if and only if R is right artinian and every indecomposable projective right R-module is uniform and weakly R-injective. We show that in the above characterization the requirement that indecomposable projective right R-modules be uniform is superfluous (Theorem 1.11). In this paper we further the study of weak relative-injectivity by considering the class of rings for which every right module is weakly injective relative to every finitely generated right module. We refer to such rings as right weakly-semisimple rings. The class of right weakly-semisimple rings includes properly all semisimple rings and is a subclass of the class of right QI-rings. A ring R is said to be a right QI-ring if every quasi-injective right R-module is injective. QI-rings have been studied in [2], [3], [4], [6], [7], [8], [11], among others.

An associative ring R is called a left SI-ring if every singular left R-module is injective. In Goodearl [4] it is shown that these rings have a finite ring decomposition into a ring K with K/Soc K left semisimple, and simple rings which are Morita equivalent to left SI-domains.

We have studied relationships between almost relative projectivity and Nakayamaings [8]. In this paper we shall further investigate certain characterizations of right Nakayama rings in terms of almost relative projectives (or injectives). We shall consider three conditions (A), (B) and (C) (see Section 1), which are always satisfied for the relative projective modules, but not for almost relative projectives in general. As an application of [9, Theorem] and [10, Theorem 2], we shall show that a right artinian ring is right Nakayama if and only if one of the above three conditions holds true for almost relative projectives (Corollary to Theorem 1). Moreover we shall give a characterization of two-sided Nakayama rings related to (C) and the dual (C#) (Theorem 2). Finally we shall investigate the transitivity of almost relative projectives, which is the converse of (B), and give some characterizations of right Nakayama rings related to the transitivity.

In [3], Sharp and Taherizadeh introduced concepts of reduction and integral closure of an ideal I of a commutative ring R (with identity) relative to an Artinian R-module A, and they showed that these concepts have properties which reflect some of those of the classical concepts of reduction and integral closure introduced by Northcott and Rees in [2].

In this note we characterize certain types of spectral decomposition in terms of “universal” notions valid for any operator on a Banach space. To be precise, let X be a complex Banach space and let T be a bounded linear operator on X. If F is a closed set in the plane C, let X(T, F) consist of all y ∈ X satisfying thes identity

where f:C\F → X is analytic. It is then easy to see that X(T, F) is a T-invariant linear manifold in X. Moreover, if y ∈ X then

is a compact subset of the spectrum σ(T). Our aim is to give necessary and sufficient conditions for a decomposable or strongly decomposable operator in terms of X(T, F) and γ(y, T). Recall that T is decomposable if whenever G1G2 are open and cover C there exist T-invariant closed linear manifolds M1, M2 with X= M1 + M2 and σ(T | M1) ⊂ Gi(i = 1,2) (equivalently, σ(T | Mi)⊂ Ḡi, see [4, p. 57]). In this case, X(T, F) is norm closed if Fis closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C\Fy; Fy is called the local spectrum σ(y, T) and coincides with γ(y, T). Hence T has the single valued extension property (SVEP); i.e., zero is the only analytic function f:V → X satisfying (z − T)f(z) = 0 on V. If T is decomposable and the restriction T | X(T, F) is also decomposable for each closed F, then T is called strongly decomposable. We point out that Albrecht [2] has shown by example that not every decomposable operator is strongly decomposable, while Eschmeier [6]has given a simpler construction to show that this phenomenon occurs even in Hilbert space.

The previous results on isometrically characterizing ln∞ in terms of operator ideal norms can be summarized as follows.

Let E be an n-dimensional Banach space.

(1) If λ(E) = 1, then (see [3], [5], [6]), where λ(E) is the projection constant of E (see [4]).

(2) If π(E) = n, then (see [1],[2]).

(3) If Δ2(E) = √n, then (see [8]), where Δ2(E) is the 2-dominated constant of E(see [4]).

(4) If for any linear operator T: → E, v1(T) = (T), then (see [7]), where v1 is the 1-nuclear norm (see [4]).

Let Hbe a complex Hilbert space and let B(H) be the algebra of (bounded) operators on H. Let A =(A,…,An) be an n-tuple of operators on H. The joint numerical range of A is the subset W(A) of ℂn such that

In the course of classifying those finite groups F which have exactly five maximal subgroups, R. W. van der Waall [4] proved that one encounters the following situation. One class of such groups F is described by F = SP, where S = O2(F)∈Syl2(F), P ∈ Syl3(F), S/Φ(S) ≅ Z2 × Z2, P is cyclic and P operates via conjugation on 5 as a group of order 3, because in this case F/Φ(F) ≅ A4.