Fixed points of asymptotically regular semigroups in Banach spaces

In this paper we study in Banach spaces the existence of fixed points of (nonlinear) asymptotically regular semigroups. We establish for these semigroups some fixed point theorems in spaces with weak uniform normal structure, in a Hilbert space, inL ^p spaces, in Hardy spacesH ^p and in Sobolev spacesW ^r.p for 1<p<∞ andr≥0, in spaces with Lifshitz’s constant greater than one. These results are the generalizations of [8, 10, 16].     

Remarks on generators of analytic semigroups

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL ^p , 1<p<∞, while the sufficient condition in the other characterization is meaningful in the case of nonlinear operators. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under Grant No. MCS78-01245.     

Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

Arens Regularity of Semigroup Algebras with Certain Locally Convex Topologies

The authors have recently introduced and studied a locally convex topology β¹(S) on the semigroup algebra M_a(S) of a locally compact semigroup S; as the main result, they showed that the strong dual of (M_a(S),β¹(S)) can be identified with the Banach space L^∞₀(S,M_a(S)) for a large class of locally compact semigroups S. Here, an application of this result is made to define and investigate an Arens multiplication on the second dual of (M_a(S),β¹(S)).     

Denting and strongly extreme points in the unit ball of spaces of operators

For 1 ≤p ≤ ∞ we show that there are no denting points in the unit ball of ℓ(l^p). This extends a result recently proved by Grząślewicz and Scherwentke whenp = 2 [GS1]. We also show that for any Banach spaceX and for any measure space (Ω, A, μ), the unit ball of ℓ(L ¹ (μ), X) has denting points iffL ¹(μ) is finite dimensional and the unit ball ofX has a denting point. We also exhibit other classes of Banach spacesX andY for which the unit ball of ℓ(X, Y) has no denting points. When X* has the extreme point intersection property, we show that all ‘nice’ operators in the unit ball of ℓ(X, Y) are strongly extreme points.     

The Singularity Property of Banach Function Spaces and Unconditional Convergence in L 1[0, 1]

Let X be a Banach function space, L^∞ [0, 1] ⊂ X ⊂ L¹[0, 1]. It is proved that if dual space of X has singularity property in closed set E ⊂ [0, 1] then: 1) there exists no orthonormal basis in C[0, 1], which forms an unconditional basis in X in metric of L¹[0, 1] space, 2) for the Hardy-Littlewood maximal operator M we have      

On set-valued cone absolutely summing maps

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L ^p (μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L ¹(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.     

A random walk for the solution sought: Remarks on the difference scheme approach to nonlinear semigroups and evolution operators

In [2], Crandall and Evans show existence of mild solution to an abstract Cauchy Problem: u′(t)+Au(t)∋f(t), 0≤t≤T, u(0)=x₀, where A is an accretive operator in a general Banach space X and f ε L¹(0,T;X). Their method involves proving convergence in the L^∞-norm of a sequence of step function approximations α_n(σ, τ) to the solution of a first order partial differential equation. We consider a more general Cauchy Problem and show a.e. existence of mild solution by proving convergence of the step functions α_n(σ, τ) in the L¹-norm. Fundamental to the proof is a nonhomogeneous random walk in the plane.     

Uniform algebras as Banach spaces

Let A be a closed subalgebra of the complex Banach algebra C(S), containing the constant functions. We assume that one has found a probability measureμ on S and a function F from L^∞(μ) such that: 1)|F|= 1 a.e. relative to μ; 2) F _μ ε A¹; 3) F is a limit point of the unit ball of the algebra A in the topology δ(L^∞(μ), L¹(μ)). One proves in the paper that under these conditions the space A** contains a complement space, isometric to H^∞. The measure μ and the function F, satisfying the conditions l)-3) indeed exist if the maximal ideal space of the algebra A contains a non-one-point part (and it is very likely that such aμ. and F exist whenever the algebra A is not self-adjoint). Thus, the above-formulated result allows us to extend A. Pelczynski's theorem (Ref, Zh. Mat., 1975, 1B894) regarding the space H^∞ to a very broad class of uniform algebras. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 80–89, 1976.     

On order structure and operators in L ∞(μ)

It is known that there is a continuous linear functional on L _∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L _∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L _∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L _∞(μ) to c ₀(Γ) is narrow while not every such an operator is AM-compact.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

Greedy algorithms and bestm-term approximation with respect to biorthogonal systems

The article extends upon previous work by Temlyakov, Konyagin, and Wojtaszczyk on comparing the error of certain greedy algorithms with that of best m-term approximation with respect to a general biorthogonal system in a Banach space X. We consider both necessary and sufficient conditions which cover most of the special cases previously considered. Some new results concerning the Haar system in L₁, L_∞, and BMO are also included.     

Lipschitz continuity of quantile functions on spaces of random variables

We prove that quantile functions on spaces of random variables satisfy the Lipschitz condition with constant 1 with respect to any norm on a subspace of a space of random variables that majorizes L^∞-norm. The considered random variables not necessarily belong to this Banach space. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351–358, 2008, pp. 253–258.     

Characterization ofL p (R n ) using Gabor frames

We characterize L^p norms of functions onR ⁿ for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L^p functions converge to the functions almost everywhere and in L^p for 1<p<∞. In L¹ we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L¹ in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L^p (R ⁿ) when 1≤p<∞.     

Irreducibility and Mixed Boundary Conditions

In this paper we consider positive semigroups on L^p(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L^∞(Ω) is real and satisfies b ≥ δ for some δ > 0. In memoriam Helmut H. Schaefer     

Existence of equilibrium for economies with infinitely many commodities and infinitely countable consumers

This paper examines the existence of equilibria for double infinite eonomies. S.F. Richard and S. Srivastava have established the existence of equilibria for economies with infinitely countable consumers when commodity space isL ^∞ (M, M, μ). However, most Banach Lattices as commodity spaces haven’t interior points in their positive cones, so their result can’t be applied to many cases. We here consider a general Banach Lattice as commodity space and introduce a concept of equiproperness on preferences. Under the assumption the existence of equilibrium for economy is established. Finally, we examine the existence of equilibria for production economies.     

Hardy spaces of vector-valued functions

The purpose of this paper is to consider the relations among the various definitions of the spaces H^p of analytic functions with values in a Banach space and to investigate the problem of the structure of the conjugates of these spaces. In particular, one constructs an example of a reflexive separable Banach space χ, for which the equality H^p(X)^*=H^p′(X^*) (1<p<∞,1/p+1/p′=1) ails. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 5–16, 1976.     

Retarded functional differential equations with nondense domain operators

In this work, we use integrated semigroups to state results on the existence and uniqueness of integral solutions and solutions for the abstract Cauchy problem x^′(t)=Bx(t)+Lx_t, t⩾0, where B is a nondensely defined linear operator on a Banach space X. This revised version was published online in June 2006 with corrections to the Cover Date.     

Compactness in Vector-valued Banach Function Spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L_X^p, where X is a Banach space and 1≤ p<∞, and extend the result to vector-valued Banach function spaces E_X, where E is a Banach function space with order continuous norm. The author is supported by the ‘VIDI subsidie’ 639.032.201 in the ‘Vernieuwingsimpuls’ programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281.