If f(z) is univalent (regular and one-to-one) in the open unit disk Δ, Δ = {z ∊ C:│z│ < 1}, and has a Maclaurin series expansion of the form

(1.1)

then, as de Branges has shown, │ak│ = k, for k = 2, 3, … and the Koebe function.

(1.1)

serves to show that these bounds are the best ones possible (see [3]). The functions defined above are generally said to constitute the class .

Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C0 and D0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D0f. In case a = 0, Murty and Rao [4] have shown that C0 × D0/f is isomorphic to L, and hence that C0 × D0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C0 × D0/f is isomorphic to L, and that C0 × D0/f is also implicative.

In recent years there has been a growing interest in problems of factorization for bounded linear operators. We first show that many of these problems properly belong to the category of C*-algebras. With this interpretation, it becomes evident that the problem is fundamental both to the structure of operator algebras and the elements therein. In this paper we consider the direct integral algebra with separable and infinite dimensional. We generalize a theorem of Wu (1988) and characterize those decomposable operators which are products of non-negative decomposable operators. We do this by first showing that various results on operator ranges may be generalized to “measurable fields of operator ranges”.

The object of this paper is to prove a version of the Beurling–Helson–Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in $\mathbb{C}$. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the closed operators that commute with all analytic multiplication operators.

In this article we study the geometry of the eigenvarieties of unitary groups at points corresponding to tempered non-stable representations with an anti-ordinary (a.k.a evil) refinement. We prove that, except in the case where the Galois representation attached to the automorphic form is a sum of characters, the eigenvariety is non-smooth at such a point, and that (under some additional hypotheses) its tangent space is big enough to account for all the relevant Selmer group. We also study the local reducibility locus at those points, proving that in general, in contrast with the case of the eigencurve, it is a proper subscheme of the fiber of the eigenvariety over the weight space.

In this paper weak Perron integrals are characterized as n-dimensional interval functions F which are additive, differentiable almost everywhere in the weak sense and which satisfy a new continuity condition concerning the singular set. Before, only one-dimensional Perron integrals were characterized by the theorem of Hake- Alexendrov-Looman, and analogous results for strong Perron integrals (which are best analyzed, but more restrictive) are not available in higher dimensions yet. In order to formulate our continuity condition we introduce an outer measure μ by means of a new weak variation of F which is required to vanish on all null sets. The same condition is also necessary and sufficient for the integral of the weak derivative to yield the original interval function. This “fundamental theorem” is split into two fundamental inequalities of very general nature which contain additional singular terms involving our variation. These inequalities are very useful also for Lebesgue integrals.

Throughout this note, an operator will always mean a bounded linear operator acting on a Hilbert space X into itself, unless otherwise stated. The class Cρ (0 < ρ < ∞ ) of operators, considered by Sz.-Nagy and Foiaş [5], is defined as follows: An operator T is in Cρ if Tnx = pPUnx for all x ∊ X, n = 1, 2, . . . , where U is a unitary operator on some Hilbert space Y containing X as a subspace, and P is the orthogonal projection of Y onto X. In [2] Holbrook defined the operator radii wρ(·) (0 < ρ ≦ ∞ ) as the generalized Minkowski distance functionals on the Banach algebra of bounded linear operators on X, i.e.,

and w∞(T) = r(T), the spectral radius of T [2, Theorem 5.1].

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.

In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

When a set of axioms is laid down as the basis of any mathematical doctrine, it must be proved that this set never leads to a contradiction. In this note we turn the question around. A set of axioms is given and we wish to adjoin an axiom of a specified type. How far does the demand of non-contradiction limit the choice of the new axiom?

We investigate here the notion of a topological ring of quotients of a topological ring with respect to an arbitrary Gabriel (idempotent) filter of right ideals. We describe the topological ring of quotients first as a subring of the algebraic ring of quotients, and then show it is a topological bicommutator of a topological injective R-module. Unlike R. L. Johnson in [6] and F. Eckstein in [2] we do not always make the ring an open subring of its ring of quotients. This would exclude examples such as C(X), the ring of continuous real-valued functions on a compact space, and its ring of quotients as described in Fine, Gillman and Lambek [3].

In a recent paper [6], P. Seibt has obtained the following result: Let k be a field of characteristic 0, k[T1, … , Tr] the polynomial ring in r indeterminates over k, and let P be a prime ideal of k[T1, … , Tr]. Then a polynomial F belongs to the n-th symbolic power P(n) of P if and only if all higher derivatives of F from the 0-th up to the (n – l)-st order belong to P.

In this work we shall naturally generalize this result so as to be valid for primes of the polynomial ring over a perfect field k. Actually, we shall get a generalization as a corollary to a theorem which asserts: For regular primes P in a k-algebra R of finite type, a certain differential filtration of R associated with P coincides with the symbolic power filtration (P(n))n≧0.

We use a special tiling for the hyperbolic d-space $\mathbb {H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$, where P is a polytope in $\mathbb {R}^d$ and $\mathcal {N}$ a net in $\mathbb {H}^d$ coming from the tiling. This implies that the spaces $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net $\mathcal {M}$ in $\mathbb {H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm {Lip}(\mathbb {H}^d)$ and $\mathrm {Lip}(\mathbb {R}^d)$.

Let $M$ be the product of two compact Hamiltonian $T$-spaces $X$ and $Y$. We present a formula for evaluating integrals on the symplectic reduction of $M$ by the diagonal $T$ action. At every regular value of the moment map for $X\,\times \,Y$, the integral is the convolution of two distributions associated to the symplectic reductions of $X$ by $T$ and of $Y$ by $T$. Several examples illustrate the computational strength of this relationship. We also prove a linear analogue which can be used to find cohomology pairings on toric orbifolds.

A sequence, a1 < a2 < a3 < …, of positive integers is called lacunary if the difference sequence dn = an+l — an tends to infinity as n → ∞.

In several recent papers we have made use of these sequences in analysis and combinatorics. In [6] we show that the class of all sets which are either finite or the range of a lacunary sequence is “full” in the sense that if (tk) is a real sequence and for each then (tk) is an l1 sequence, that is,

In [3] the class of all finite unions of sets of is shown to consist of exactly those sets of integers, A, whose characteristic sequence, χA, is in the well known summability space bs + c0. More recently, in [1], we study lacunary sequences in connection with the conjecture of P.

In this paper we study sharp localized ${{L}^{q}}\,\to \,{{L}^{p}}$ estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on 2×2 matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling–Ahlfors operator.

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).

A theorem of Eisenbud, Griffith, and Robson states that if R is hereditary and noetherian (on both the left and right) then every proper homomorphic image of R is a generalized unserial ring (see, for example, [3, p. 244]). Singh [11, p. 883] states a converse: If R is a right bounded, noetherian prime ring, all of whose proper homomorphic images are generalized uniserial rings, then (every divisible right R-module is injective, so) R is right hereditary. (Actually, Singh omitted the clearly necessary “bounded” condition.) Singh's theorem generalizes results of [9, Proposition 15], [2, Theorem 2.1], and [8], about commutative rings.

We will call a semi-prime ring R essentially right bounded if each essential right ideal contains a two-sided ideal which is essential as a right ideal. In case R is prime, “essentially right bounded” coincides with “right bounded”.

Let $L$ be an oriented Lagrangian submanifold in an $n$-dimensional Kähler manifold $M$. Let $u:\,D\,\to \,M$ be a minimal immersion from a disk $D$ with $u(\partial D)\,\subset \,L$ such that $u(D)$ meets $L$ orthogonally along $u(\partial D)$. Then the real dimension of the space of admissible holomorphic variations is at least $n\,+\,\mu (E,\,F)$, where $\mu (E,\,F)$ is a boundary Maslov index; the minimal disk is holomorphic if there exist $n$ admissible holomorphic variations that are linearly independent over $\mathbb{R}$ at some point $p\,\in \,\partial D;$; if $M=\mathbb{C}{{P}^{n}}$ and $u$ intersects $L$ positively, then $u$ is holomorphic if it is stable, and its Morse index is at least $n\,+\,\mu (E,\,F)$ if $u$ is unstable.

On post-critically finite self-similar sets, whose walk dimensions of diffusions are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces $B^{p,q}_\sigma (K)$ and the Lipschitz–Besov spaces $\Lambda ^{p,q}_\sigma (K)$, are identical. In particular, we provide concrete examples that $B^{p,q}_\sigma (K)=\Lambda ^{p,q}_\sigma (K)$ with $\sigma>1$. Our method is purely analytical, and does not involve heat kernel estimate.

For almost any compact connected Lie group $G$ and any field $\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra $H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$ on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if $p$ is odd or $p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology $HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over $\mathbb{F}_{2}$ , such an isomorphism of Batalin–Vilkovisky algebras does not hold when $G=\text{SO}(3)$ or $G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.