In this paper we show that the normal parts of quasisimilar log-hyponormal operators are unitarily equivalent. A Fuglede-Putnam type theorem for log-hyponormal operators is proved. Also, it is shown that a log-hyponormal operator that is quasisimilar to an isometry is unitary and that a log-hyponormal spectral operator is normal.

An ordered semigroup S will be called principally ordered if, for every x ɛ S, there exists

x* = max {y ɛ S; xyx ≤ x}.

Here we shall be concerned with the case where S is regular. We begin by listing some basic properties that arise from the above definition. As usual, we shall denote by V(x) the set of inverses of x ɛ S.

This paper discusses the fractional chromatic number of the direct product of graphs. It is proved that if H is a circulant graph G^k_d, or a Kneser graph, or a direct sum of such graphs, then for any graph G, \chi_f{\hskip1}(G\times H{\hskip1}) = {\text min}\{\chi_f{\hskip1}(G), \chi_f{\hskip1}(H{\hskip1})\}.

Let ℋ be a complex Hilbert space and B(ℋ) the algebra of all bounded linear operators on ℋ. Let ℋ(ℋ) be the algebra of all compact operators of B(ℋ). For an operator T ε B(ℋ), let σ(T), σp(T), σπ(T) and πoo(T) denote the spectrum, the point spectrum, the approximate point spectrum and the set of all isolated eigenvalues of finite multiplicity of T, respectively. We denote the kernel and the range of an operator T by ker(T) and R(T), respectively. For a subset of ℋ, the norm closure of is denoted by . The Weyl spectrum ω(T) of T ε B(ℋ) is defined as the set

A kernel functor (equivalently, a left exact torsion preradical) is a left exact subfunctor of the identity on the category R-mod of left R-modules over a ring R with identity. A kernel functor is said to be idempotent if, in addition, σ satisfies σ(M / σ(M)) = 0 for every M ∊ R-mod. To every kernel functor / there corresponds a unique topologizing filter ℒσ = {I Ⅰ σ (R/I) = R/I} of left ideals and a unique class ℱσ = {M ∊ R-mod Ⅰ σ(M) = M} that is closed under homomorphic images, submodules, and direct sums. The idempotence of σ is characterized by either of the following additional conditions:

(1) if I ∊ ℒσ, K ⊆ I, and (K:x) = {r ∊ R ∣ rx ∊ K} ∊ ℒσ for each x ∊ I, then K ∊ ℒ or

(2) ℱσ is closed under extensions of one member of ℱσ by another member of ℱσ Idempotent kernel functors are important since they are the tool used to construct localization functors. For M∊ R-mod, let E(M) denote the injective hull of M. A kernel functor σ is called stable if Mℱ implies that E(M) ∊ ℱσ For more information about kernel functors, see [6], [7], [14], and [15].

Given a normed linear space X, let S(X), X′, B(X) denote respectively the unit sphere {x: ∥x∥ = 1} of X, the dual space of X, and the algebra of all bounded linear mappings of X into X. For each x ∊ S(X) and T ∊ B(X), let Dx(x) = {f e X′:∥f∥ = f(x)= 1}, and V(T; x) = {f(Tx):f∊Dx(x)}. The numerical range V(T) is then defined by

We obtain estimates for the degree of bilipschitz determinacy of quasihomogeneous function-germs.

The main object of study in this paper is the quantized Weyl algebra which arises from the work of Maltsiniotis [10] on noncommutative differential calculus. This algebra has been studied from the point of view of noncommutative ring theory by various authors including Alev and Dumas [1], the second author [9], Cauchon [3], and Goodearl and Lenagan [5]. In [9], it is shown that has n normal elements zi and, subject to a condition on the parameters, the localization obtained on inverting these elements is simple of Krull and global dimension n. It is easy to show that each of these normal elements generates a height one prime ideal and that these are all the height one prime ideals of . The purpose of this paper is to determine, under a stronger condition on the parameters, all the prime ideals of and to compare the prime spectrum with that of a related algebra . This algebra has more symmetric defining relations than those of but it shares the same simple localization which again is obtained by inverting n normal elements zi. Like the alternative algebra can be regarded as an algebra of skew differential (or difference) operators on the coordinate ring of quantum n-space.

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.

In [1], we showed how a collection of physical operations or experiments could be represented by a nonempty set of nonempty sets satisfying certain conditions (irredundancy and coherence) and we called such sets . We also introduced “complete stochastic models” for the empirical universe of discourse represented by such a manual , namely, the so-called weight functions for . These weight functions form a convex set the extreme points of which are called pure weights. We also showed that there is a so-called logic ∏() affiliated with a manual and that each weight function for induces a state on this logic.

The following Chen's bi-harmonic conjecture made in 1991 is well-known and stays open: The only bi-harmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we prove that the bi-harmonic conjecture is true for bi-harmonic hypersurfaces with three distinct principal curvatures of a Euclidean space of arbitrary dimension.

Let X be a Banach space and L(X) the Banach algebra of bounded linear operators on X. An operator T in L(X) is hermitian if ∥eitT∥ = 1 (t ∈ R), and is normal if T = R + iJ where R and J are commuting normal operators; R and J are then determined uniquely by T, and we may write T* = R–iJ. These definitions extend those for operators on Hilbert spaces. More details may be found in [1].

A ring $R$ is called left morphic if, for every $a\in R$, $R/Ra\cong {\bf l}(a)$ where ${\bf l}(a)$ denotes the left annihilator of $a$ in $R$. Right morphic rings are defined analogously. In this paper, we investigate when the trivial extension $R\propto M$ of a ring $R$ and a bimodule $M$ over $R$ is (left) morphic. Several new families of (left) morphic rings are identified through the construction of trivial extensions. For example, it is shown here that if $R$ is strongly regular or semisimple, then $R\propto R$ is morphic; for an integer $n>1$, ${\mathbb Z}_n\propto {\mathbb Z}_n$ is morphic if and only if $n$ is a product of distinct prime numbers; if $R$ is a principal ideal domain with classical quotient ring $Q$, then the trivial extension $R\propto {Q}/{R}$ is morphic; for a bimodule $M$ over $\mathbb Z$, ${\mathbb Z}\propto M$ is morphic if and only if $M\cong{\mathbb Q}/{\mathbb Z}$. Thus, ${\mathbb Z}\propto {\mathbb Q}/{\mathbb Z}$ is a morphic ring which is not clean. This example settled two questions both in the negative raised by Nicholson and Sánchez Campos, and by Nicholson, respectively.

The problem considered is the determination of “lower bounds” of matrix operators on the spaces\ell_p(w) or d(w,p). Under fairly general conditions, the solution is the same for both spaces and is given by the infimum of a certain sequence. Specific cases are considered, with the weighting sequence defined by w_n = 1/n^\alpha . The exact solution is found for the Hilbert operator. For the averaging operator, two different upper bounds are given, and for certain values of p and \alpha it is shown that the smaller of these two bounds is the exact solution.

We find the lower bound for the norm of the Hilbert matrix operator H on the weighted Bergman space Ap,α

\begin{equation*} \|H\|_{A^{p,\alpha}\rightarrow A^{p,\alpha}}\geq\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}, \,\, \textnormal{for} \,\, 1<\alpha+2<p. \end{equation*}

$\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}$

We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets $\mathcal{A}$ ⊂ ℤ and show quantitative bounds on the size of $\mathcal{A}$ if there are no non-trivial solutions. We use the circle method and Roth's density increment argument. Due to a restriction theory approach we can deal with equations in s ≥ 7 variables.

For any positive integers n and v let

where d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,

as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

In this paper we generalize the formula of Frobenius-Stickelberger (see (0.1) below) and the formula of Kiepert (see (0.2) below) to the genus-two case.

We consider a special class of solutions of the BKP hierarchy which we call $\tau$-functions of hypergeometric type. These are series in Schur $Q$-functions over partitions, with coefficients parameterised by a function of one variable $\xi$, where the quantities $\xi(k)$, $k\in\mathbb{Z^+}$, are integrals of motion of the BKP hierarchy. We show that this solution is, at the same time, a infinite soliton solution of a dual BKP hierarchy, where the variables $\xi(k)$ are now related to BKP higher times. In particular, rational solutions of the BKP hierarchy are related to (finite) multi-soliton solution of the dual BKP hierarchy. The momenta of the solitons are given by the parts of partitions in the Schur $Q$-function expansion of the $\tau$-function of hypergeometric type. We also show that the KdV and the NLS soliton $\tau$-functions coinside the BKP $\tau$-functions of hypergeometric type, evaluated at special point of BKP higher time; the variables $\xi$ (which are BKP integrals of motions) being related to KdV and NLS higher times.

We prove that every infinite subset of the dual of a compact, connected group contains an infinite, central, weighted I0 set. This yields a new proof of the fact that the duals of such groups admit infinite central p-Sidon sets for each p > 1. We also establish the existence of infinite, weighted I0 sets in the duals of many compact, abelian hypergroups.