Generic Fr\'echet-differentiability and perturbed optimization problems in Banach spaces
Ivar
Ekeland;
Gérard
Lebourg
193-216
Abstract: We define a function F on a Banach space V to be locally $\varepsilon$-supported by ${u^\ast} \in {V^\ast}$ at $u \in V$ if there exists an $\eta > 0$ such that $\left\Vert {v - u} \right\Vert \leqslant \eta \Rightarrow F(v) \geqslant F(u) + \langle {u^\ast},v - u\rangle - \varepsilon \left\Vert {v - u} \right\Vert$. We prove that if the Banach space V admits a nonnegative Fréchet-differentiable function with bounded nonempty support, then, for any $> 0$ and every lower semicontinuous function F, there is a dense set of points $u \in V$ at which F is locally $\varepsilon$-supported. The applications are twofold. First, to the study of functions defined as pointwise infima; we prove for instance that every concave continuous function defined on a Banach space with Fréchet-differentiable norm is Fréchet-differentiable generically (i.e. on a countable intersection of open dense subsets). Then, to the study of optimization problems depending on a parameter $u \in V$; we give general conditions, mainly in the framework of uniformly convex Banach spaces with uniformly convex dual, under which such problems generically have a single optimal solution, depending continuously on the parameter and satisfying a first-order necessary condition.
A decomposition of weighted translation operators
Joseph J.
Bastian
217-230
Abstract: Let (X, A, m) be a measure space and T an invertible measure-preserving transformation on X. Given $\phi$ in $ {L^\infty }(X)$, define operators ${M_\phi }$ and ${U_T}$ on ${L^2}(X)$ by $({M_\phi }f)(x) = \phi (x)f(x)$ and $({U_T}f) = f(Tx)$. Operators of the form ${M_\phi }{U_T}$ are called weighted translation operators. In this paper it is shown that every weighted translation operator on a sufficiently regular measure space an be decomposed into a direct integral of weighted translation operators where almost all of the transformations in the integrand are ergodic. It is also shown that every hyponormal weighted translation operator defined by an ergodic transformation is either normal or unitarily equivalent to a bilateral weighted shift. These two results along with some results concerning direct integrals of hyponormal and subnormal operators are used to show that every hyponormal (resp. subnormal) weighted translation operator is unitarily equivalent to a direct integral of normal operators and hyponormal (resp. subnormal) bilateral weighted shifts. The paper concludes with an example.
Absolute Tauberian constants for Ces\`aro means of a function
Soraya
Sherif
231-242
Abstract: This paper is concerned with introducing two estimates of the forms $F \leqslant C{A_k}(\alpha ),F \leqslant D{B_k}(\alpha ),(\alpha > 0)$, where $F = \smallint_0^\infty {\vert d\{ f(\alpha x) - {\sigma _k}(x)\} \vert,{\sigma _k}(x)}$ denote the Cesàro transform of order k of the function $ f(x) = \smallint_0^x {g(t)\;dt,g(t)}$ is a function of bounded variation in every finite interval of $t \geqslant 0,{A_k}(\alpha ),{B_k}(\alpha )$ are absolute Tauberian constants, $C = \smallint_0^\infty {\vert d\{ tg(t)\} \vert < \infty ,D = \smallint_0^\infty {\vert d\{ \phi (t)\} \vert < \infty } }$ and $\phi (t) = {t^{ - 1}}\smallint_0^t {ug(u)du}$. The constants ${A_k}(\alpha ),{B_k}(\alpha )$ will be determined.
Fringe families in stable homotopy
Raphael S.
Zahler
243-253
Abstract: It is shown how to detect and construct elements in the stable homotopy groups of spheres corresponding to the $ {_i}$ family of Toda. The only tools used are Brown-Peterson cohomology and the Adams spectral sequence.
Coefficient multipliers of Bloch functions
J. M.
Anderson;
A. L.
Shields
255-265
Abstract: The class $\mathcal{B}$ of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by $c/(1 - r)$ on $\vert z\vert \leqslant r$. We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from ${l^p}$ into $ \mathcal{B}$ and from $\mathcal{B}$ into ${l^p}$. We find the second Köthe dual of $\mathcal{B}$ and show its relevance to the multiplier problem. We identify all power series $\sum {a_n}{z^n}$ such that $\sum {w_n}{a_n}{z^n}$ is a Bloch function for every choice of the bounded sequence $\{ {w_n}\}$. Analogous problems for ${H^p}$ spaces are discussed briefly.
Unique factorization in modules and symmetric algebras
Douglas L.
Costa
267-280
Abstract: Necessary and sufficient conditions are given for a torsion-free module M over a UFD D to admit a smallest factorial module containing it. This factorial hull is $\cap {M_P}$, the intersection taken over all height one primes of D. In case M is finitely generated, the hull is ${M^{ \ast \ast }}$, the bidual of M. It is shown that if the symmetric algebra $ {S_D}(M)$ admits a hull, then the hull is the smallest graded UFD containing $ {S_D}(M)$. ${S_D}(M)$ is a UFD if and only if it is a factorial D-module. If M is finitely generated over D, but not necessarily torsion-free, then ${ \oplus _{i \geqslant 0}}{({S^i}(M))^{ \ast \ast }}$ is a graded UFD. Examples are given to show that any finite number of symmetric powers of M may be factorial without ${S_D}(M)$ being factorial.
A minimax formula for dual $B\sp*$-algebras
Pak Ken
Wong
281-298
Abstract: Let A be a dual ${B^\ast}$-algebra. We give a minimax formula for the positive elements in A. By using this formula and some of its consequent results, we introduce and study the symmetric norms and symmetrically-normed ideals in A.
The Cartesian product structure and $C\sp{\infty }$ equivalances of singularities
Robert
Ephraim
299-311
Abstract: In this paper the cartesian product structure of complex analytic singularities is studied. A singularity is called indecomposable if it cannot be written as the cartesian product of two singularities of lower dimension. It is shown that there is an essentially unique way to write any reduced irreducible singularity as a cartesian product of indecomposable singularities. This result is applied to give an explicit description of the set of reduced irreducible complex singularities having a given underlying real analytic structure.
Duality theory for locally compact groups with precompact conjugacy classes. II. The dual space
Terje
Sund
313-321
Abstract: The present paper is concerned with the dual space Ĝ consisting of all unitary equivalence classes of continuous irreducible unitary representations of separable ${[FC]^ - }$ groups (i.e. groups with precompact conjugacy classes). The main purpose of the paper is to extend certain results from the duality theory of abelian groups and [Z] groups to the larger class of ${[FC]^ - }$ groups. In addition, we deal briefly with square-integrability for representations of ${[FC]^ - }$ groups. Most of our results are proved for type I groups. Our key result is that Ĝ may be written as a disjoint union of abelian topological $ {T_4}$ groups, which are open in Ĝ.
Continuously perfectly normal spaces and some generalizations
Gary
Gruenhage
323-338
Abstract: In this work we continue the study of continuously perfectly normal, continuously normal, and continuously completely regular spaces which was begun by Phillip Zenor. Among other results, we prove that separable continuously completely regular spaces are metrizable, and provide an example of a nonmetrizable continuously perfectly normal space.
Indecomposable homogeneous plane continua are hereditarily indecomposable
Charles L.
Hagopian
339-350
Abstract: F. Burton Jones [7] proved that every decomposable homogeneous plane continuum is either a simple closed curve or a circle of homogeneous nonseparating plane continua. Recently the author [5] showed that no subcontinuum of an indecomposable homogeneous plane continuum is hereditarily decomposable. It follows from these results that every homogeneous plane continuum that has a hereditarily decomposable subcontinuum is a simple closed curve. In this paper we prove that no subcontinuum of an indecomposable homogeneous plane continuum is decomposable. Consequently every homogeneous nonseparating plane continuum is hereditarily indecomposable. Parts of our proof follow one of R. H. Bing's arguments [2]. At the Auburn Topology Conference in 1969, Professor Jones [8] outlined an argument for this theorem and stated that the details would be supplied later. However, those details have not appeared.
A reduction theory for non-self-adjoint operator algebras
E. A.
Azoff;
C. K.
Fong;
F.
Gilfeather
351-366
Abstract: It is shown that every strongly closed algebra of operators acting on a separable Hilbert space can be expressed as a direct integral of irreducible algebras. In particular, every reductive algebra is the direct integral of transitive algebras. This decomposition is used to study the relationship between the transitive and reductive algebra problems. The final section of the paper shows how to view direct integrals of algebras as measurable algebra-valued functions.
Analysis with weak trace ideals and the number of bound states of Schr\"odinger operators
Barry
Simon
367-380
Abstract: We discuss interpolation theory for the operator ideals $ I_p^w$ defined on a separable Hilbert space as those operators A whose singular values ${\mu _n}(A)$ obey ${\mu _n} \leqslant c{n^{ - 1/p}}$ for some c. As an application we consider the functional $N(V) = \dim$ (spectral projection on $( - \infty ,0)$ for $ - \Delta + V$) on functions V on ${{\mathbf{R}}^n},n \geqslant 3$. We prove that for any $ \epsilon > 0:N(V) \leqslant C_\epsilon (\left\Vert V \right\Vert _{n/2 + \epsilon } + \left\Vert V \right\Vert _{n/2 - \epsilon})^{n/2}$ where ${\left\Vert \cdot \right\Vert _p}$ is an $ {L^p}$ norm and that $ {\lim\nolimits _{\lambda \to \infty }}N(\lambda V)/{\lambda ^{n/2}} = {(2\pi )^{ - n}}{\tau _n}\smallint \vert{V_ - }(x){\vert^{n/2}}{d^n}x$ for any $V \in {L^{n/2 - }} \cap {L^{n/2 + }}$. Here $ {V_ - }$ is the negative part of V and ${\tau _n}$ is the volume of the unit ball in ${{\mathbf{R}}^n}$.
Natural limits for harmonic and superharmonic functions
J. R.
Diederich
381-397
Abstract: In this paper it is shown that Fatou's theorem holds for superharmonic functions in certain Liapunov domains if mean continuous limits are used in place of nontangential limits for which Fatou's theorem fails. Also, existence of mean continuous limits is established for certain semi-linear elliptic equations in Liapunov domains.
Nonimmersion of lens spaces with 2-torsion
A. J.
Berrick
399-405
Abstract: From a study of the equivariant unitary K-theory of the Stiefel manifold $ {V_{k + 1,2}}({\mathbf{C}})$, it is shown that the lens space ${L^k}(n)$, with n a multiple of $ {2^{2k - 1 - \alpha (k - 1)}}$, does not immerse in Euclidean space of dimension $4k - 2\alpha (k) - 2$.
Quasi-similar models for nilpotent operators
C.
Apostol;
R. G.
Douglas;
C.
Foiaş
407-415
Abstract: Every nilpotent operator on a complex Hilbert space is shown to be quasi-similar to a canonical Jordan model. Further, the para-reflexive operators are characterized generalizing a result of Deddens and Fillmore.
The degree of approximation for generalized polynomials with integral coefficients
M.
von Golitschek
417-425
Abstract: The classcal Müntz theorem and the so-called Jackson-Müntz theorems concern uniform approximation on [0, 1] by polynomials whose exponents are taken from an increasing sequence of positive real numbers $\Lambda$. Under mild restrictions on the exponents, the degree of approximation for $ \Lambda$-polynomials with real coefficients is compared with the corresponding degree of approximation when the coefficients are taken from the integers.
Deformation of open embeddings of $Q$-manifolds
A.
Fathi;
Y. M.
Visetti
427-435
Abstract: We prove here for Hilbert cube manifolds the full analogue of the Černavski-Edwards-Kirby Theorem concerning the deformation Principle for open embeddings of topological manifolds.
Some consequences of the algebraic nature of $p(e\sp{i\theta })$
J. R.
Quine
437-442
Abstract: For polynomial p of degree n, the curve $p({e^{i\theta }})$ is a closed curve in the complex plane. We show that the image of this curve is a subset of an algebraic curve of degree 2n. Using Bézout's theorem and taking into account imaginary intersections at infinity, we show that if p and q are polynomials of degree m and n respectively, then the curves $p({e^{i\theta }})$ and $q({e^{i\theta }})$ intersect at most 2mn times. Finally, let ${U_k}$ be the set of points w, not on $p({e^{i\theta }})$, such that $ p(z) - w$ has exactly k roots in $\vert z\vert < 1$. We prove that if L is a line then $L \cap {U_k}$ has at most $n - k + 1$ components in L and in particular ${U_n}$ is convex.