Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups

The theory of symmetric local semigroups due to A. Klein and L. Landau (J. Funct. Anal.44 (1981), 121–136) is generalized to semigroups indexed by subsets of $\text{R}$ⁿ for n > 1. The result implies a similar result of A. E. Nussbaum (J. Funct. Anal.48 (1982), 213–223). It is further generalized to semigroups that are symmetric local in some directions and unitary in others. The results are used to give a simple proof of A. Devinatz's (Duke Math. J.22 (1955), 185–192) and N. I. Akhiezer's (“the Classical Moment Problem and Some Related Questions,” Hafner, New York, 1965) generalization of a theorem of Widder concerning the representation of functions as Laplace integrals. This result is extended to the representation as a Laplace integral of a function taking values in $\text{B}$($\text{R}$), the set of bounded linear operators on a Hilbert space $\text{R}$. Also, a theorem is proved encompassing both the result of Devinatz and Akhiezer, and Bochner's theorem on the representation of positive definite functions as Fourier integrals.     

Left mean-ergodicity,fixed points,and invariant means

Recently Lau [15] generalized a result of Yeadon [25]. In the present paper we generalize Yeadon's result in another direction recasting it as a theorem of ergodic type. We call the notion of ergodicity required left mean-ergodicity and show how it relates to the mean-ergodicity of Nagel [21]. Connections with the existence of invariant means on spaces of continuous functions on semitopological semigroups S are made, connections concerning, among other things, a fixed point theorem of Mitchell [20] and Schwartz's property P of W¹-algebras [22]. For example, if $\text{M}$(S) is a certain subspace of C(S) (which was considered by Mitchell and is of almost periodic type, i.e., the right translates of a member of $\text{M}$(S) satisfy a compactness condition), then the assumption that $\text{M}$(S) has a left invariant mean is equivalent to the assumption that every representation of S of a certain kind by operators on a linear topological space X is left mean-ergodic. An analog involving the existence of a (left and right) invariant mean on $\text{M}$(S) is given, and we show our methods restrict in the Banach space setting to give short direct proofs of some results in [4], results involving the existence of an invariant mean on the weakly almost periodic functions on S or on the almost periodic functions on S. An ergodic theorem of Lloyd [16] is generalized, and a number of examples are presented.     

Some fractional differintegral equations

Recently, Nishimoto [J. College. Engrg. Nihon Univ. Ser. B24 (1983), 7–13] studied a certain fractional integral equation and a system of simultaneous differintegral equations, each of order $\text{1}\text{2}$. Motivated by Nishimoto's work, Owa and Nishimoto [J. College Engrg. Nihon Univ. Ser. B24 (1983), 67–72] considered a general fractional differintegral equation. The object of the present paper is to investigate some interesting properties of functions which satisfy the general fractional differintegral equation solved by Owa and Nishimoto.     

A fixed point theorem for eventually nonexpansive semigroups of mappings

Let K be a subset of a Banach space X. A semigroup $\text{F}$ = {?_α ∥ α ∞ A} of Lipschitz mappings of K into itself is called eventually nonexpansive if the family of corresponding Lipschitz constants $\text{{k}{}_{\mathrm{\alpha }}\text{¦ α ? A}}$ satisfies the following condition: for every ? > 0, there is a γ?A such that k_β < 1 + ? whenever ?_β ? ?_γ$\text{F}$ = {?_gg?_α ¦ ?_α ? $\text{F}$}. It is shown that if K is a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space, and if $\text{F}$:K → K is an eventually nonexpansive, commutative, linearly ordered semigroup of mappings, then $\text{F}$ has a common fixed point. This result generalizes a fixed point theorem by Goebel and Kirk.     

Projection theorems for hitting probabilities and a theorem of Littlewood

Littlewood (Proc. London Math. Soc. (2), 28 1928, 383–394) showed that a positive superharmonic function u on the unit disc has radial limits a.e. Using techniques due to Doob this result is extended to all rank one symmetric spaces. In addition simplifications are obtained of Doob's (Ann. Inst. Fourier (Grenoble), 15 1965, 113–135) proof of normal convergence a.e. of a positive superharmonic function on a half space. The symmetric space analogue of this half space result is also obtained. The methods used are shown to fail for the potential theory on $\text{R}$ⁿ associated with Δu = αu (α > 4 0). It is an open question as to whether Littlewood's theorem holds in this context.     

Paley-Wiener theorem for singular support of K-finite distributions on symmetric spaces

We study Fourier transforms of distributions on a symmetric space X. Eguchi et al. [1] characterized the image of $\text{E}$′(X)-distributions of compact support under the Fourier transform. We give a simpler proof of Eguchi's result and characterize the size of the singular support for the K-finite members of E′(X). We apply this Paley-Wiener type theorem to invariant differential equations on X.     

The Todd character and cohomology operations

In “On the Conflict of Bordism of Finite Complexes” [J. Differential Geometry], Conner and Smith introduced a homomorphism called the Todd character, relating complex bordism theory to rational homology. Specifically the Todd character consists of a family of homomorphisms

$\text{th}{}_{\mathrm{r}}\text{: MU}{}_{\mathrm{s}}\text{(X) → H}{}_{\mathrm{s\to r}}\text{(X;}\text{Q}\text{)}$

.In L. Smith, The Todd character and the integrality theorem for the Chern character, Ill. J. Math. it was shown (note that the indexing of the Todd character is somewhat different here) that there was an integrality theorem for th analogous to the Adams integrality theorem for the Chern character J. F. Adams, On the Chern character and the structure of the unitary group, Proc. Cambridge Philos. Soc.57 (1961), 189–199; On the Chern character revisted, Ill. J. Math. Now Adams' first paper contains a wealth of information about the Chern character in addition to the integrality theorem already mentioned. Our objective in the present note is to derive analogous results for the Todd character. As in Smith these may then be used to deduce the results of Adams for the Chern character.     

A note on GL2 converse theoremsUne note sur les théorèmes inverses de GL2

Weil's well-known converse theorem shows that modular forms $\text{f∈}\text{M}{}_{\mathrm{k}}\text{(Γ}{}_0\text{(q))}$ are characterized by the functional equation for twists of L_f(s). Conrey–Farmer had partial success at replacing the assumption on twists by the assumption of L_f(s) having an Euler product of the appropriate form. In this Note we obtain a hybrid version of Weil's and Conrey–Farmer's results, by proving a converse theorem for all q?1 under the assumption of the Euler product and, moreover, of the functional equation for the twists to a single modulus. To cite this article: A. Diaconu et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 621–624.     

On splitting the area under curves II

This paper continues the author's work [3, S. Minsker, J. Differential Equations, 26, No. 3 (1977), 443–457.] on an area-splitting problem leading to the functional differential equation $\text{a′(a(x)) =}\text{a(x)}\text{x}$. This equation is dealt with by transforming it into the linear equation ψ′(x) = ψ(x + c), for which positive solutions on (?∞, ?c) are sought.     

Strong ergodic theorem for commutative semigroup of non-Lipschitzian mappings in multi-Banach space

Let C be a bounded closed convex subset of a uniformly convex multi-Banach space X and let \({\mathfrak {I}}_{j} = \{T_j(t) : t\in G\}\) be a commutative semigroup of asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we prove the strong mean ergodic convergence theorem for the almost-orbit of \(\mathfrak {I}\). Our results extend and unify many previously known results especially (Dong et al. On the strong ergodic theorem for commutative semigroup of non-Lipschitzian mappings in Banach space, preprint).     

Cohomologie des groupes localement compacts et produits tensoriels continus de représentations

The following is an expository paper, containing few and sometimes incomplete proofs, on continuous tensor products of Hilbert spaces and of group representations, and on the irreducibility of the latter; the principal results in the last direction are due to Verchik, Gelfand, and Graiev. The theory of continuous tensor products of Hilbert spaces, based on a fundamental theorem of Araki and Woods, is closely related to that of conditionally positive definite functions; it relies on the technique of symmetric Hilbert spaces, which also can be used to give a new proof of the classical Lévy-Khinchin formula (see A. Guichardet, (1973). J. Multiv.3 249–261.). Another basic tool for what follows is the 1-cohomology of unitary representations of locally compact groups; here, the main results are due to P. Delorme; let us mention, for instance, his results for the case of a group G containing a compact subgroup K such that $\text{L}{}^1\text{(Kβ}\text{G}\text{K}\text{)}$ is commutative, using a Lévy-Khinchin's type formula for K-invariant functions due to Gangolli, Faraut, and Harzallah. We add that the results exposed in that paper should have interesting connections with the central limit theorems à la Parthasarathy-Schmidt (see K. Parthasarathy, (1974). J. Multiv. Anal.4 123–149).     

On finite differences,permutations, and Quadratic residues

Let p be an odd prime and n an integer relatively prime to p. In this work three criteria which give the value of the Legendre symbol ($\text{n}\text{p}$) are developed. The first uses two adjacent rows of Pascal's triangle which depend only on p to express ($\text{n}\text{p}$) explicitly in terms of the numerically least residues (mod p) of the numbers n, 2n, …, [$\text{(p + 1)}\text{4}\text{]n}$ or of the numbers $\text{[}\text{(p + 1)}\text{4}\text{]n,…, [}\text{(p ? 1)}\text{2}\text{]n}$. The second, analogous to a theorem of Zolotareff and valid only if p ≡ 1 (mod 4), expresses ($\text{n}\text{p}$) in terms of the parity of the permutation of the set {$\text{1,2,…, (}\text{(p? 1)}\text{2}$} defined by the absolute values of the numerically least residues of $\text{n, 2n,…,[}\text{(p? 1}\text{2}\text{]n}$. The third is a result dual to Gauss' lemma which can be derived directly without Euler's criterion. The applications of the dual include a proof of Gauss' lemma free of Euler's criterion and a proof of the Quadratic Reciprocity Law.     

Beilinson resolutions on weighted projective spaces

Beilinson's theorem [Funct. Anal. Appl. 12 (1978) 214–216], which describes the bounded derived category of coherent sheaves on $\text{P}{}^{\mathrm{n}}$, is extended to weighted projective spaces. This result is obtained by considering, instead of the usual category of coherent sheaves, a suitable category of graded coherent sheaves (which is equivalent in the case of $\text{P}{}^{\mathrm{n}}$). To cite this article: A. Canonaco, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Strong convergence theorems by hybrid method for asymptotically -strict pseudo-contractive mapping in Hilbert space

In this paper, we introduce the hybrid method of modified Mann’s iteration for an asymptotically k-strict pseudo-contractive mapping. Then we prove that such a sequence converges strongly to P_F(T)x₀. This main theorem improves the result of Issara Inchan [I. Inchan, Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces, Int. J. Math. Anal. 2 (23) (2008) 1135–1145] and concerns the result of Takahashi et al. [W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 341 (2008) 276–286], and many others.     

Projections of polynomial hulls

The following theorem is discussed. Let X be a compact subset of the unit sphere in $\text{C}$ⁿ whose polynomially convex hull, $\text{X}\text{?}$, contains the origin, then the sum of the areas of the n coordinate projections of $\text{X}\text{?}$ is bounded below by π. This applies, in particular, when $\text{X}\text{?}$ is a one-dimensional analytic subvariety V containing the origin, and in this case generalizes the fact that the “area” of V is at least π; in fact, the area of V is the sum of the areas of the n coordinate projections when these areas are counted with multiplicity. A convex analog of the theorem is obtained. Hartog's theorem that separate analyticity implies analyticity, usually proved with the use of subharmonic functions (Hartog's lemma), will be derived as a consequence of the theorem, the proof of which is based upon the elements of uniform algebras.     

A generalization of Turán's theorem to directed graphs

We consider an extremal problem for directed graphs which is closely related to Turán's theorem giving the maximum number of edges in a graph on n vertices which does not contain a complete subgraph on m vertices. For an integer n?2, let T_n denote the transitive tournament with vertex set X_n={1,2,3,…,n} and edge set {(i,j):1?i<j?n}. A subgraph H of T_n is said to be m-locally unipathic when the restriction of H to each m element subset of X_n consisting of m consecutive integers is unipathic. We show that the maximum number of edges in a m-locally unipathic subgraph of T_n is ($\text{q}\text{2}\text{)(m?1)}{}^2\text{+q(m?1)r+?}\text{1}\text{4}\text{r}{}^2\text{?}$ where n= q(m?1+r and $\text{?}\text{1}\text{2}\text{(m?1)??r<?}\text{3}\text{2}\text{(m?1)?}$. As is the case with Turán's theorem, the extremal graphs for our problem are complete multipartite graphs. Unlike Turán's theorem, the part sizes will not be uniform. The proof of our principal theorem rests on a combinatorial theory originally developed to investigate the rank of partially ordered sets.     

On the existence of uncountably many matroidal families

A matroidal family is a set $\text{F}$ ≠ ? of connected finite graphs such that for every finite graph G the edge-sets of those subgraphs of G which are isomorphic to some element of $\text{F}$ are the circuits of a matroid on the edge-set of G. Simões-Pereira [5] shows the existence of four matroidal families and Andreae [1] shows the existence of a countably infinite series of matroidal families. In this paper we show that there exist uncountably many matroidal families. This is done by using an extension of Andreae's theorem, a construction theorem, and certain properties of regular graphs. Moreover we observe that all matroidal families so far known can be obtained in a unified way.     

Grothendieck's theorem for noncommutative C1-algebras,with an Appendix on Grothendieck's constants

We study a conjecture of Grothendieck on bilinear forms on a C¹-algebra $\text{Ol}$. We prove that every “approximable” operator from $\text{Ol}$ into $\text{Ol}$¹ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C¹-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant k_G.