Spectral families of projections, semigroups, and differential operators
Harold
Benzinger;
Earl
Berkson;
T. A.
Gillespie
431-475
Abstract: This paper presents new developments in abstract spectral theory suitable for treating classical differential and translation operators. The methods are specifically geared to conditional convergence such as arises in Fourier expansions and in Fourier inversion in general. The underlying notions are spectral family of projections and well-bounded operator, due to D. R. Smart and J. R. Ringrose. The theory of well-bounded operators is considerably expanded by the introduction of a class of operators with a suitable polar decomposition. These operators, called polar operators, have a canonical polar decomposition, are free from restrictions on their spectra (in contrast to well-bounded operators), and lend themselves to semigroup considerations. In particular, a generalization to arbitrary Banach spaces of Stone's theorem for unitary groups is obtained. The functional calculus for well-bounded operators with spectra in a nonclosed arc is used to study closed, densely defined operators with a well-bounded resolvent. Such an operator $L$ is represented as an integral with respect to the spectral family of its resolvent, and a sufficient condition is given for $(- L)$ to generate a strongly continuous semigroup. This approach is applied to a large class of ordinary differential operators. It is shown that this class contains significant subclasses of operators which have a polar resolvent or generate strongly continuous semigroups. Some of the latter consist of polar operators up to perturbation by a semigroup continuous in the uniform operator topology.
The number of factorizations of numbers less than $x$ into factors less than $y$
Douglas
Hensley
477-496
Abstract: Let $K(x,y)$ be the number in the title. There is a function $f(r)$, concave and decreasing with $ f(0) = 2$ and $f^{\prime}(0) = 0$ such that if $r = \sqrt {\log x} /\log y$ then as $x \to \infty$ with $r$ fixed, $\displaystyle K(x,y) = x \exp\,\left({f(r)\,\sqrt {\log x} + O\,{{(\log \log x)}^2}} \right)$ . The proof uses a uniform version of Chernoff's theorem on large deviations from the sample mean of a sum of $ N$ independent random variables.
Generic algebras
John
Isbell
497-510
Abstract: The familiar (merely) generic algebras in a variety $\mathcal{V}$ are those which separate all the different operations of $ \mathcal{V}$, or equivalently lie in no proper Birkhoff subcategory. Stronger notions are considered, the strongest being canonicalness of a (small) subcategory $\mathcal{A}$ of $ \mathcal{V}$, defined: the structure functor takes inclusion $\mathcal{A} \subset \mathcal{V}$ to an isomorphism of varietal theories. Intermediate are dominance and exemplariness: lying in no proper varietal subcategory, respectively full subcategory. It is shown that, modulo measurable cardinals, every finitary variety has a canonical set (subcategory) of one or two algebras, the possible second one being the empty algebra. Without reservation, every variety with rank has a dominant set of one or two algebras (the second as before). Finally, in modules over a ring $R$, the generic module $R$ is shown to be (a) dominant if exemplary, and (b) dominant if $R$ is countable or right artinian. However, power series rings $R$ and some others are not dominant $ R$-modules.
An algebraic approach to Grothendieck's residue symbol
Glenn
Hopkins
511-537
Abstract: A certain map--the "residue map"--is defined and its properties are investigated. The impetus for the definition and study of this map is a definition by A. Grothendieck of a homomorphism, the "residue symbol", which has been found applicable in several areas, including the duality theory of algebraic varieties.
Computability and noncomputability in classical analysis
Marian Boykan
Pour-El;
Ian
Richards
539-560
Abstract: This paper treats in a systematic way the following question: which basic constructions in real and complex analysis lead from the computable to the noncomputable, and which do not? The topics treated include: computability for $ {C^n}$, ${C^\infty }$, real analytic functions, Fourier series, and Fourier transforms. A final section presents a more general approach via "translation invariant operators". Particular attention is paid to those processes which occur in physical applications. The approach to computability is via the standard notion of recursive function from mathematical logic.
Sufficient conditions for the generalized problem of Bolza
Vera
Zeidan
561-586
Abstract: This paper presents sufficient conditions for strong local optimality in the generalized problem of Bolza. These conditions represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data. Also, this paper brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations, showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original Hamiltonian to a locally concave-convex Hamiltonian.
Skewness in Banach spaces
Simon
Fitzpatrick;
Bruce
Reznick
587-597
Abstract: Let $E$ be a Banach space. One often wants to measure how far $E$ is from being a Hilbert space. In this paper we define the skewness $s(E)$ of a Banach space $E$, $0 \leqslant s(E) \leqslant 2$, which describes the asymmetry of the norm. We show that $s(E) = s({E^{\ast}})$ for all Banach spaces $ E$. Further, $ s(E) = 0$ if and only if $ E$ is a (real) Hilbert space and $s(E) = 2$ if and only if $E$ is quadrate, so $s(E) < 2$ implies $E$ is reflexive. We discuss the computation of $ s({L^p})$ and describe its asymptotic behavior near $p = 1,2$ and $\infty$. Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.
Pseudojump operators. I. The r.e. case
Carl G.
Jockusch;
Richard A.
Shore
599-609
Abstract: Call an operator $ J$ on the power set of $ \omega$ a pseudo jump operator if $J(A)$ is uniformly recursively enumerable in $ A$ and $A$ is recursive in $J(A)$ for all subsets $A$ of $\omega$. Thus the (Turing) jump operator is a pseudo jump operator, and any existence proof in the theory of r.e. degrees yields, when relativized, one or more pseudo jump operators. Extending well-known results about the jump, we show that for any pseudo jump operator $J$, every degree $\geqslant {\mathbf{0}}^{\prime}$ has a representative in the range of $J$, and that there is a nonrecursive r.e. set $A$ with $J(A)$ of degree ${\mathbf{0}}^{\prime}$. The latter result yields a finite injury proof in two steps that there is an incomplete high r.e. degree, and by iteration analogous results for other levels of the ${H_n}$, ${L_n}$ hierarchy of r.e. degrees. We also establish a result on pairs of pseudo jump operators. This is combined with Lachlan's result on the impossibility of combining splitting and density for r.e. degrees to yield a new proof of Harrington's result that ${\mathbf{0}}^{\prime}$ does not split over all lower r.e. degrees.
An embedding characterization of almost compact spaces
Sergio
Salbany
611-621
Abstract: We characterize almost compact and almost realcompact spaces in terms of their situation in the product ${(J,u)^C}$. In the characterization of almost compactness $J$ is the two point set or the unit interval; in the characterization of almost realcompactness $ J$ is the set of nonnegative integers or the nonnegative reals. $ u$ is the upper topology on the real line restricted to $J$.
$L\sp{p}$ multipliers with weight $x\sp{kp-1}$
Benjamin
Muckenhoupt;
Wo Sang
Young
623-639
Abstract: Let $k$ be a positive integer and $1 < p < \infty$. It is shown that if $ T$ is a multiplier operator on ${L^p}$ of the line with weight $\vert x{\vert^{kp-1}}$, then $Tf$ equals a constant times $ f$ almost everywhere. This does not extend to the periodic case since $m(j) = 1/j, j \ne 0$, is a multiplier sequence for $ {L^p}$ of the circle with weight $ \vert x{\vert^{kp-1}}$. A necessary and sufficient condition is derived for a sequence $m(j)$ to be a multiplier on ${L^2}$ of the circle with weight $\vert x{\vert^{2k - 1}}$.
Schr\"odinger operators with rapidly oscillating central potentials
Denis A. W.
White
641-677
Abstract: Spectral and scattering theory is discussed for the Schrödinger operators $H = - \Delta + V$ and ${H_0} = - \Delta$ when the potential $V$ is central and may be rapidly oscillating and unbounded. A spectral representation for $ H$ is obtained along with the spectral properties of $H$. The existence and completeness of the modified wave operators is also demonstrated. Then a condition on $V$ is derived which is both necessary and sufficient for the Møller wave operators to exist and be complete. This last result disproves a recent conjecture of Mochizuki and Uchiyama.
Regular functions of restricted growth and their zeros in tangential regions
C. N.
Linden
679-686
Abstract: For a given function $k$, positive, continuous, nondecreasing and unbounded on $[0,1)$, let ${A^{(k)}}$ denote the class of functions regular in the unit disc for which log $\vert f(z)\vert < k(\vert z\vert)$ when $ \vert z\vert < 1$. Hayman and Korenblum have shown that a necessary and sufficient condition for the sets of positive zeros of all functions in ${A^{(k)}}$ to be Blaschke is that $\displaystyle \int_0^1 {\sqrt {(k(t)/(1 - t))\,dt} }$ is finite. It is shown that the imposition of a further regularity condition on the growth of $k$ ensures that in some tangential region the zero set of each function in ${A^{(k)}}$ is also Blaschke.
Dehn surgery and satellite knots
C. McA.
Gordon
687-708
Abstract: For certain kinds of $3$-manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in ${S^3}$ is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain $ {S^3}$, a fake $ {S^3}$, a fake $ {S^3}$ with nonzero Rohlin invariant, $ {S^1} \times {S^2}$, a fake ${S^1} \times {S^2}, {S^1} \times {S^2} \char93 M$ with $M$ nonsimply-connected, or a fake lens space.
AF algebras with directed sets of finite-dimensional $\sp{\ast} $-subalgebras
Aldo J.
Lazar
709-721
Abstract: We characterize the unital $AF$ algebras whose families of finite dimensional $^{\ast}$-subalgebras are directed by inclusion. A representation theorem for the algebras of this class allows us to classify them up to $^{\ast} $-isomorphisms.
Characterizations of simply connected rotationally symmetric manifolds
Hyeong In
Choi
723-727
Abstract: We prove that a simply connected, complete Riemannian manifold $ M$ is rotationally symmetric at $p$ if and only if the exponential image of every linear subspace of ${M_p}$ is a smooth, closed, totally geodesic submanifold of $M$. This result is in essence Schur's theorem at one point $p$, as it becomes apparent in the proof.
Differential group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions
Reinhard
Schultz
729-750
Abstract: A linear action of an abelian group on a sphere generally contains a large family of invariant linear subspheres. In this paper the problem of finding invariant subspheres for more general smooth actions on homotopy spheres is considered. Classification schemes for actions with invariant subspheres are obtained; these are formally parallel to the classifications discussed in the preceding paper of this series. The realizability of a given smooth action as an invariant codimension two subsphere is shown to depend only on the ambient differential structure and an isotopy invariant. Applications of these results to specific cases are given; for example, it is shown that every exotic $10$-sphere admits a smooth circle action.
Brownian motion and a generalised little Picard's theorem
Wilfrid S.
Kendall
751-760
Abstract: Goldberg, Ishihara, and Petridis have proved a generalised little Picard's theorem for harmonic maps; if a harmonic map of bounded dilatation maps euclidean space, for example, into a space of negative sectional curvatures bounded away from zero then that map is constant. In this paper a probabilistic proof is given of a variation on this result, requiring in addition that the image space has curvatures bounded below. The method involves comparing asymptotic properties of Brownian motion with the asymptotic behaviour of its image under such a map.
Measurable representations of preference orders
R. Daniel
Mauldin
761-769
Abstract: A continuous preference order on a topological space $Y$ is a binary relation $\preccurlyeq$ which is reflexive, transitive and complete and such that for each $x,\{y:x \preccurlyeq y\} $ and $\{y:y \preccurlyeq x\}$ are closed. Let $ T$ and $X$ be complete separable metric spaces. For each $t$ in $T$, let ${B_t}$ be a nonempty subset of $X$, let ${ \preccurlyeq _t}$ be a continuous preference order on ${B_t}$ and suppose $E = \{(t,x,y): x{ \preccurlyeq _t}y\}$ is a Borel set. Let $B = \{(t,x):x \in {B_t}\} $. Theorem 1. There is an $\mathcal{S}(T) \otimes \mathcal{B}(X)$-measurable map $g$ from $B$ into $R$ so that for each $t,g(t,\cdot)$ is a continuous map of $ {B_t}$ into $ R$ and $g(t,x) \leqslant g(t,y)$ if and only if $x{ \preccurlyeq _t}y$. (Here $\mathcal{S}(T)$ forms the $C$-sets of Selivanovskii and $\mathcal{B}(X)$ is a Borel field on $ X$.) Theorem 2. If for each $t,{B_t}$ is a $\sigma$-compact subset of $Y$, then the map $g$ of the preceding theorem may be chosen to be Borel measurable. The following improvement of a theorem of Wesley is proved using classical methods. Theorem 3. Let $ g$ be the map constructed in Theorem 1. If $\mu$ is a probability measure defined on the Borel subsets of $T$, then there is a Borel set $N$ such that $\mu (N) = 0$ and such that the restriction of $ g$ to $B \cap ((T - N) \times X)$ is Borel measurable.
Stochastic waves
E. B.
Dynkin;
R. J.
Vanderbei
771-779
Abstract: Let $\phi$ be a real valued function defined on the state space of a Markov process $ {x_t}$. Let ${\tau _t}$ be the first time $ {x_t}$ gets to a level set of $\phi$ which is $t$ units higher than the one on which it started. We call the time changed process $\tilde{x}_{t} = x_{{\tau_t}}$ a stochastic wave. We give conditions under which this process is Markovian and we evaluate its infinitesimal operator.
Weighted norm inequalities for homogeneous families of operators
José L.
Rubio de Francia
781-790
Abstract: If a family of operators in ${R^n}$ is invariant under rotations and dilations and satisfy a certain inequality in ${L^p}({l^r})$, then it is uniformly bounded in the weighted space ${L^r}(\vert x\vert{^{n(r/p - 1)}}\,dx)$. This is the main consequence of a more general result for operators in homogeneous spaces. Applications are given to certain maximal operators, the Fourier transform and Bochner-Riesz multipliers.
Structural stability and group cohomology
Philip J.
Fleming
791-809
Abstract: We prove a version of the theorem of Stowe concerning the stability of stationary points of a differentiable group action which is valid on Hilbert manifolds. This result is then used to show that the vanishing of certain cohomology groups is sufficient to guarantee structural semistability for a differentiable action of a group of finite type on a closed smooth manifold. We then apply this to groups of diffeomorphisms of the circle.
A sphere theorem for manifolds of positive Ricci curvature
Katsuhiro
Shiohama
811-819
Abstract: Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that $ {\text{Ri}}{{\text{c}}_M} \geqslant 1,{K_M} \geqslant - {\kappa ^2}$ and the volume of $M \geqslant$ the volume of the $(\pi - \varepsilon )$-ball on the unit $ m$-sphere, $m = {\text{dim }}M$. Then for a suitable choice of $ \varepsilon = \varepsilon (m,k)$ every $M$ belonging to this class is homeomorphic to $ {S^m}$.
A general maximal operator and the $A\sb{p}$-condition
M. A.
Leckband;
C. J.
Neugebauer
821-831
Abstract: A rearrangement inequality for a general maximal operator $Mf(x) = {\sup _{x \in Q}}\int {f\phi_{Q}\,d\nu }$ is established. This is then applied to the Hardy-Littlewood maximal operator with weights.
Krull dimension of differential operator rings. III. Noncommutative coefficients
K. R.
Goodearl;
T. H.
Lenagan
833-859
Abstract: This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring $S[\theta ;\delta ]$, where $S$ is a right noetherian ring with finite Krull dimension $n$ and $\delta$ is a derivation on $S$. The main theorem states that $S[\theta ;\delta ]$ has Krull dimension $ n$ unless there exists a simple right $S$-module $A$ such that $A{ \otimes _S}S[\theta ;\delta ]$ is not simple (as an $ S[\theta ;\delta ]$-module) and $A$ has height $n$ in the sense that there exist critical right $ S$-modules $A = {A_0},{A_1},\ldots,{A_n}$ such that each ${A_i} \otimes_s S[\theta ;\delta ]$ is a critical $S[\theta ;\delta ]$-module, each $ {A_i}$ is a minor subfactor of ${A_{i + 1}}$ and ${A_n}$ is a subfactor of $S$. If such an $A$ does exist, then $S[\theta ;\delta ]$ has Krull dimension $ n + 1$. This criterion is simplified when $S$ is fully bounded, in which case it is shown that $S[\theta ;\delta ]$ has Krull dimension $n$ unless $S$ has a maximal ideal $M$ of height $n$ such that either ${\text{char(}}S/M) > 0$ or $\delta (M) \subseteq M$, and in these cases $S[\theta ;\delta ]$ has Krull dimension $ n + 1$.