Uniqueness of Minimal Projections in Smooth Matrix Spaces

Let X=(M(n, m), ·), where · fulfills Condition 0.3 and W=M(n, 1)+M(1, m). A formula for a minimal projection from X onto W is given in (E. W. Cheney and W. A. Light, 1985, “Approximation Theory in Tensor Product Spaces,” Lecture Notes in Mathematics, Springer-Verlag, Berlin; E. J. Halton and W. A. Light, 1985, Math. Proc. Cambridge Philos. Soc.97, 127–136; and W. A. Light, 1986, Math. Z.191, 633–643). We will show that this projection is the unique minimal projection (see Theorem 2.1).     

Linear functionals on nonlinear spaces and applications to problems from viscoplasticity theory

A classical result in the theory of monotone operators states that if C is a reflexive Banach space, and an operator A: C→C^* is monotone, semicontinuous and coercive, then A is surjective. In this paper, we define the ‘dual space’ C^* of a convex, usually not linear, subset C of some Banach space X (in general, we will have C^*⊃X^*) and prove an analogous result. Then, we give an application to problems from viscoplasticity theory, where the natural space to look for solutions is not linear. Copyright © 2007 John Wiley & Sons, Ltd.     

Approximation numbers for composition operators on spaces of entire functions

We study the degree of compactness of composition operators $C_{\phi }$ acting on weighted Hilbert spaces of entire functions, which include (i) the space of entire Dirichlet series, (ii) the space of entire power series, and (iii) the Fock space (we must have $\phi \left(z\right)=az+b$, and it is known that $C_{\phi }$ is compact if and only if $\left|a\right|<1$). More precisely, the sequence $\left(a_n\right)$ of approximation numbers of $C_{\phi }$ is investigated: for (i), we give the exact formula for $\left(a_n\right)$, while for (ii) and (iii) we give upper and lower estimates for $a_n$, showing that $a_n$ behaves like ${\left|a\right|}^n$ up to a subexponential factor. In particular, $C_{\phi }$ belongs to all Schatten classes $S_p,p>0$ as soon as it is compact.     

On Uniformly Finitely Extensible Banach spaces

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno and Plichko (2009) [39] and Castillo and Plichko (2010) [18]. We show that they have the Uniform Approximation Property of Pe?czyński and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal – do there exist automorphic spaces other than _c0(I)$c_0(I)$ and _?2(I)${?}_2(I)$? – showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI – among them, the super-reflexive HI space constructed by Ferenczi – and asymptotically _?2${?}_2$ spaces in the literature cannot be automorphic.     

Compactness of Bochner representable operators on Orlicz spaces

We study compactness properties of linear operators from an Orlicz space L^Φ provided with a natural mixed topology to a Banach space (X, || · ||_X). We derive that every Bochner representable operator is -compact. In particular, it is shown that every Bochner representable operator is (τ(L^∞, L¹), || · ||_X)-compact.      

Disjointly strictly singular inclusions of symmetric spaces

In this paper, the disjoint strict singularity of inclusions of symmetric spaces of functions on an interval is considered. A condition for the presence of a “gap” between spaces sufficient for the inclusion of one of these spaces into the other to be disjointly strictly singular is found. The condition is stated in terms of fundamental functions of spaces and is exact in a certain sense. In parallel, necessary and sufficient conditions for an inclusion of Lorentz spaces to be disjointly strictly singular (and similar conditions for Marcinkiewicz spaces) are obtained and certain other assertions are proved. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 3–14, January, 1999.     

On the convergence properties of sampling Durrmeyer-type operators in Orlicz spaces

Here, we provide a unifying treatment of the convergence of a general form of sampling-type operators, given by the so-called sampling Durrmeyer-type series. The main result consists of the study of a modular convergence theorem in the general setting of Orlicz spaces $L^{\phi }(\mathbb{R})$. From the latter theorem, the convergence in $L^p(\mathbb{R})$, in $L^{\alpha }{\log }^{\beta }L$, and in the exponential spaces can be obtained as particular cases. For the completeness of the theory, we provide a pointwise and uniform convergence theorem on $\mathbb{R}$, in case of bounded continuous and bounded uniformly continuous functions; in this context, we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.     

On a generalization of the extended best polynomial approximation operator in Orlicz–Lorentz spaces

In this paper, we consider the best polynomial approximation operator defined on an Orlicz–Lorentz space ${\mathrm{\Lambda }}_{w,\varphi }$, and its extension to ${\mathrm{\Lambda }}_{w,{\varphi }^{\prime }}$, where w is a non-negative continuous weight function and ${\varphi }^{\prime }$ is the derivative of ϕ, which is not required to be an Orlicz function. Our work generalizes a recent result in this field on an Orlicz–Lorentz space generated by an Orlicz function. In addition, we establish some properties and estimates for any extended best polynomial approximation.     

Noncommutative orlicz-hardy spaces

Let (Φ, Ψ) be a pair of complementary N-functions and HΦ(A)$H^{\Phi }\left(A\right)$ and HΨ(A)$H^{\Psi }\left(A\right)$ be the associated noncommutative Orlicz-Hardy spaces. We extend the Riesz, Szegö and inner-outer type factorization theorems of Hp(A)$H^p\left(A\right)$ to this case.     

On decompositions of Banach spaces of continuous functions on Mrówka's spaces

It is well known that if is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space of continuous functions on has complemented copies of , i.e., . We address the question if this could be the only type of decompositions of into infinite-dimensional summands for infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin's axiom we construct an example of Mrówka's space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.

     

Superposition operators between weighted spaces of analytic functions

Abstract

We study the existence and the continuity of superposition operators between weighted spaces of holomorphic functions in terms of the weights.     

Nonexpansive selections of metric projections in spaces of continuous functions

A subset A of a metric space X is said to be a nonexpansive proximinal retract (NPR) of X if the metric projection from X to A admits a nonexpansive selection. We study the structure of NPR's in the space C(K) of continuous functions on a compact Hausdorff space K. The main results are a characterization of finite-codimensional and of finite-dimensional NPR subspaces of C(K) and a complete characterization of all NPR subsets of .     

Absolutely L – Summing Norms of Diagonal Operators in lr and Limit Orders of Lexp – Summing Operators

We compute the absolutely L – summing norms of the diagonal operators acting on l_r (1 ≤ q, r < ∞) and determine the limit orders of the absolutely L_exp – summing operators.     

Taylor's inequalities in Orlicz–Sobolev type spaces

In this paper, we obtain inequalities involving the Taylor polynomial and weak derivatives of a function in an Orlicz–Sobolev type space. Moreover, we show that any such function can be expanded in a finite Taylor series almost everywhere. As a consequence, we prove that the coefficients of any extended best polynomial $L^{\mathrm{\Phi }}$-approximation of a function on a ball almost everywhere converge to the weak derivatives of such a function when the radius tends to 0. Lastly, we get a mean convergence result of such coefficients.     

The asymptotically commuting bounded approximation property of Banach spaces

We introduce and study the asymptotically commuting bounded approximation property of Banach spaces. This property is, e.g., enjoyed by any dual space with the bounded approximation property. The principal result is the following: if a Banach space X has the asymptotically λ-commuting bounded approximation property, then X is saturated with locally λ-complemented separable subspaces enjoying the λ-commuting bounded approximation property.     

Complementably universal Banach spaces, II

The two main results are:

A.

If a Banach space X is complementably universal for all subspaces of c₀ which have the bounded approximation property, then X^∗ is non-separable (and hence X does not embed into c₀).

B.

There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.

Theorem B solves a problem that dates from the 1970s.     

Generalized projections, decompositions, and the Pythagorean-type theorem in Banach spaces

In this paper, we investigate new properties of the generalized projection operators on convex closed cones in uniformly convex and uniformly smooth Banach spaces; establish decompositions theorems for arbitrary elements both in primary and dual spaces; and prove the Banach space analogue of the Pythagorean-type theorem. Earlier, all these results were known only in Hilbert spaces.