Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces M^p{\mathcal{M}^{p}} and Stepanoff spaces S^p{\mathcal{S}^p}, 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (M^p₀{M^{p}_{0}}). We construct a function f that belongs to these spaces and has the property that all approximate identities f_e * f{\phi_\varepsilon * f} converge to f pointwise but they never converge in norm.     

Polynomials and identities on real Banach spaces

We study the duality theory for real polynomials and functions on Banach spaces. Our approach leads to a unified treatment and generalization of some classical results on linear identities and polynomial characterizations due to Fréchet, Mazur, Orlicz, Reznick, Wilson, and others.     

Bases in spaces of homogeneous polynomials on Banach spaces

In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces.     

Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

Approximate identities in variable L p spaces

We give conditions for the convergence of approximate identities, both pointwise and in norm, in variable L ^p spaces. We unify and extend results due to Diening [8], Samko [18] and Sharapudinov [19]. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable L ^p spaces. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Notes on Fourier series in homogeneous Banach spaces

В первой части работы устанавливается при нцип двойственности для с ходимости операторо в свертки в некоторых ф ункциональных прост ранствах. Вторая часть содержи т несколько характеристик после довательностей коэф фициентов Фурье функций из этих пространств.     

Bicircular projections on some Banach spaces

In this paper we show that the bicircular projections are precisely the Hermitian projections and prove some immediate consequences of this result.     

On perfectly homogeneous bases in Banach spaces

It is proved that a Banach space is isomorphic toc _o or tol _p if and only if it has a normalized basis {χ_i i _{} i=1} ^∞ which is equivalent to every normalized block-basis with respect to {χ_i i _{} i=1} ^∞ . This is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful guidance and for the interest he showed in the paper, and the referee for his valuable remakrs.     

Approximate identities and Young type inequalities in Musielak-Orlicz spaces

We discuss the convergence of approximate identities in Musielak-Orlicz spaces extending the results given by Cruz-Uribe and Fiorenza (2007) and the authors F.-Y. Maeda, Y. Mizuta and T. Ohno (2010). As in these papers, we treat the case where the approximate identity is of potential type and the case where the approximate identity is defined by a function of compact support. We also give a Young type inequality for convolution with respect to the norm in Musielak-Orlicz spaces.     

Multipliers on homogeneous Banach spaces with respect to Jacobi polynomials

A bounded linear operator is called multiplier with respect to Jacobi polynomials if and only if it commutes with all Jacobi translation operators on $[-1,1]$ . Multipliers on homogeneous Banach spaces on $[-1,1]$ determined by the Jacobi translation operator are introduced and studied. First we prove four equivalent characterizations of a multiplier for an arbitrary homogeneous Banach spaces $B$ on $[-1,1]$ . One of them implies the existence of an algebra isomorphism from the set of all multipliers on $B$ into the set of all pseudomeasures. Further, we study multipliers on specific examples of homogeneous Banach spaces on $[-1,1]$ . Amongst others, multipliers on the Wiener algebra, on the Beurling space and on Sobolev spaces are analyzed. We obtain that the multiplier spaces of the Wiener algebra, the Beurling space and of all Sobolev spaces are isometric isomorphic to each other. Furthermore, these multiplier spaces are all isometric isomorphic to the set of all pseudomeasures.     

Embeddings of some classical Banach spaces into modulation spaces

We give sufficient conditions for a tempered distribution to belong to certain modulation spaces by showing embeddings of some Besov-Triebel-Lizorkin spaces into modulation spaces. As a consequence we have a new proof that the Hölder-Lipschitz space embeds into the modulation space when d$">. This embedding plays an important role in interpreting recent modulation space approaches to pseudodifferential operators.

     

Cohomology automorphisms of some homogeneous spaces

We study the group of cohomology automorphisms of homogeneous spaces with positive Euler characteristics and prove a conjecture posed by Glover and Mislin. Our method is based on the analysis of the Lie algebra of the automorphism group and the properties of invariant polynomials of Weyl group.     

Approximate weak invariance for semilinear differential inclusions in Banach spaces

In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C ₀-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + [`(co)]overline {co} F(x(t)), without any Lipschitz conditions on the multi-function F.     

Numerical boundaries for some classical Banach spaces

Globevnik gave the definition of boundary for a subspace . This is a subset of Ω that is a norming set for . We introduce the concept of numerical boundary. For a Banach space X, a subset BΠ(X) is a numerical boundary for a subspace if the numerical radius of f is the supremum of the modulus of all the evaluations of f at B, for every f in . We give examples of numerical boundaries for the complex spaces X=c₀, and d_*(w,1), the predual of the Lorentz sequence space d(w,1). In all these cases (if K is infinite) we show that there are closed and disjoint numerical boundaries for the space of the functions from B_X to X which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary. In the case of c₀, we characterize the numerical boundaries for that space of holomorphic functions.     

Intrinsic characterizations of perturbation classes on some Banach spaces

We investigate relationships between inessential operators and improjective operators acting between Banach spaces X and Y, emphasizing the case in which one of the spaces is a C(K) space. We show that they coincide in many cases, but they are different in the case X = Y = C(K ₀), where K ₀ is a compact space constructed by Koszmider.     

The retraction constant in some Banach spaces

We give new sharper estimations for the retraction constant in some Banach spaces.