Grzegorz Lewicki

Symmetric spaces with maximal projection constants

In this paper we introduce a special class of finite-dimensional symmetric subspaces of L1, so-called regular symmetric subspaces. Using this notion, we show that for any k⩾2, there exist k-dimensional symmetric subspaces of L1 which have maximal projection constant among all k-dimensional symmetric spaces. Moreover, L1 is a maximal overspace for these spaces (see Theorems 4.4 and 4.5.) Also a new asymptotic lower bound for projection constants of symmetric spaces is obtained (see Theorem 5.3). This result answers the question posed in [12, p. 36] (see also [15, p. 38]) by H. Koenig and co-authors. The above results are presented both in real and complex cases.

Approximation of functions of finite variation by superpositions of a sigmoidal function

Abstract The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝ d ƒ (〈t, x〉)dμ(t) , where μ is a finite Radon measure on ℝ d and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L 2 -norm by elements of the set G n = {Σ i=0 staggeredn c i g(〈a i , x〉 + b i ) : a i ℝ d , b i , c i ℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n 1/2 , where C is a positive constant depending only on f . The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝ d ‖t‖e d |u(t)| > ∞

Chalmers--Metcalf operator and uniqueness of minimal projections

We know that not all minimal projections in are unique (see [B. Shekhtman, L. Skrzypek, On the non-uniqueness of minimal projections in Lp spaces]). The aim of this paper is examine the connection of the Chalmers?Metcalf operator (introduced in [B.L. Chalmers, F.T. Metcalf, A characterization and equations for minimal projections and extensions, J. Oper. Theory 32 (1994) 31?46]) to the uniqueness of minimal projections. The main theorem of this paper is Theorem 2.2. It relates uniqueness of minimal projections to the invertibility of the Chalmers?Metcalf operator. It is worth mentioning that to a given minimal projection (even unique) we may find many different Chalmers?Metcalf operators, some of them invertible, some not?see Example 2.6. The main application is in Section 3, where we have proven that minimal projections onto symmetric subspaces in smooth Banach spaces are unique (Theorem 3.2). This leads (in Section 4) to the solution of the problem of uniqueness of classical Rademacher projections in Lp[0,1] for 1

Best approximants in modular function spaces

Abstract In this paper we consider the existence of best approximants in modular function spaces by elements of sublattices. Modular function spaces are the natural generalization of Lp, p > 0, Orlicz, Lorentz, and Kothe spaces. Let ϱ be a pseudomodular, Lϱ the corresponding modular function space, and C a sublattice of Lϱ. Given a function f ∈ Lϱ we consider the minimization problem of finding h ∈ C such that ϱ(f − h) = inf{f − g: g ∈ C}. Such an h is called a best approximant. Problems of finding best approximants are important in approximation theory and probability theory. In the case where C is Lϱ( B ) for some σ-subalgebra B of the original σ-algebra, finding best approximants is closely related to the problem of nonlinear prediction. Throughout most of the paper we assume only that ϱ is a pseudomodular and except in one section, we do not assume ϱ to be orthogonally additive. This allows, for instance, application to Lorentz type Lp spaces. If ϱ is a semimodular or a modular, then Lϱ can be equipped with an F-norm ∥ · ∥ϱ and one considers the corresponding F-norm minimization problem. This paper gives several existence theorems relating to this problem, a theorem comparing the set of all best ϱ-approximants with the set of all best ∥ · ∥ϱ-approximants and a uniqueness theorem.

One-complemented subspaces of Musielak--Orlicz sequence spaces

The aim of this paper is to characterize one-complemented subspaces of finite codimension in the Musielak-Orlicz sequence space l Φ . We generalize the well-known fact (Ann. Mat. Pura Appl. 152 (1988) 53; Period. Math. Hungar. 22 (1991) 161; Classical Banach Spaces I, Springer, Berlin, 1977) that a subspace of finite codimension in l p , 1 ≤ p n ) we prove a similar characterization in l Φ . In the case of Orlicz spaces we obtain a complete characterization of one-complemented subspaces of finite codimension, which extends and completes the results in Randrianantoanina (Results Math. 33(1-2) (1998) 139). Further, we show that the well-known fact that a one-complemented subspace of finite codimension in l p , 1 ≤ p p -spaces, 1 2 -spaces, in terms of one-complemented hyperplanes, in the class of Musielak-Orlicz and Orlicz spaces as well.

Extreme and Smooth Points in Lorentz and Marcinkiewicz Spaces with Applications to Contractive Projections

We characterize extreme and smooth points in Lorentz sequence space d(w, 1) and in Marcinkiewicz sequence spaces d∗(w, 1) and d∗(w, 1), which are predual and dual spaces to d(w, 1), respectively. We then apply these characterizations for studying the relationship between the existence and one-complemented subspaces in d(w, 1). We show that a subspace of d(w, 1) is an existence set if and only if it is one-complemented. Marcinkiewicz and Lorentz spaces play an important role in the theory of Banach spaces. They are key objects for instance in the interpolation theory of linear operators. The origins of the Marcinkiewicz spaces go back to the theorem on weak type operators [23, th. 2.b.15], which was originally due to K. Marcinkiewicz in the 1930-ties. The Lorentz spaces introduced by G.G. Lorentz in 1950, have appeared in a natural way as interpolation spaces between suitable Lebesgue spaces by classical result of Lions and Peetre [23, th. 2.g.18]. This theory has been developed very extensively thereafter and along with these investigations, the theory of Lorentz and Marcinkiewicz spaces, including the studies of their geometric structure, have been evolved independently (e.g. [6, 7, 22, 25]). One can observe that these spaces find also applications in other topics of operator theory. It is worth to mention that Marcinkiewicz spaces d∗(w, 1) have emerged recently many times in the context of norm-attaining linear operators. In the papers [1, 9, 14] it was shown among others, by using the space d∗(w, 1) with specific weight, that the subspace of norm attaining operators is not always dense in the space of all bounded operators, contrary to the Bishop-Phelps theorem for linear functionals. For such types of isometric results the knowledge of geometric properties of the ball is of the utmost importance (see e.g. [9], where the characterization of complex convexity of the Lorentz spaces was the key factor in the proof of the main result). In this paper we consider the Lorentz and Marcinkiewicz sequence spaces generated by decreasing weight sequences. In the first two sections we shall characterize the smooth and extreme points of the balls in these spaces. In the last section we shall apply these results to study the relationship between the existence and one-complemented subspaces of Lorentz sequence spaces. Let’s first agree on basic definitions and notations. Throughout the paper any vector space will be always considered over the field of real numbers R. Given a Banach space (X, ‖ · ‖), by SX and BX we denote the unit sphere and the unit ball of X, respectively. Recall that x ∈ SX is an extreme Department of Mathematics, POSTECH, San 31, Hyoja-dong, Nam-gu, Pohang-shi, Kyungbuk, Republic of Korea Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA Department of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland 1 2 HAN JU LEE1,ANNA KAMIŃSKA2 AND GRZEGORZ LEWICKI3 point of the ball BX whenever x = (x1 + x2)/2 with xi ∈ SX , i = 1, 2, implies that x = x1 = x2. An element x ∈ X is called a smooth point of X if there exists a unique bounded linear functional φ ∈ SX∗ such that φ(x) = ‖x‖. Such functional φ is a called a supporting functional of x. A symbol extC will stand for the set of all extreme points of a convex subset C of X. Assume that {w(n)} is a decreasing sequence of positive numbers such that limn w(n) = 0 and ∑∞ n=1 w(n) = ∞. Let W (n) = ∑n i=1 w(i). By cardA we denote cardinality of A ⊂ N. For a real sequence x = {x(n)}, by x∗ = {x∗(n)} we denote its decreasing rearrangement. Recall that x∗(n) = inf{s > 0 : dx(s) ≤ n}, n ∈ N, where dx is a distribution of x, that is dx(s) = card{k ∈ N : |x(k)| > s}, s ≥ 0. For any x = {x(n)} the support of x is the set suppx = {n ∈ N : x(n) 6= 0}. We say that two sequences are equimeasurable whenever their distributions coincide. The Lorentz sequence space d(w, 1) is a collection of all real sequences x = {x(n)} such that

Bernstein's “lethargy” theorem in metrizable topological linear spaces

A version of Bernsteins “lethargy” theorem is given in a class of vector spaces includingF-spaces and Orlicz-Musielak spaces with condition Δ2. This extends results obtained in [9] and [11].

Codimension-one minimal projections onto Haar subspaces

Let Hn be an n-dimensional Haar subspace of X = CR[a,b] and let Hn-1 be a Haar subspace of Hn of dimension n-1. In this note we show (Theorem 6) that if the norm of a minimal projection from Hn onto Hn-1 is greater than 1, then this projection is an interpolating projection. This is a surprising result in comparison with Cheney and Morris (J. Reine Angew. Math. 270 (1974) 61 (see also (Lecture Notes in Mathematics, Vol. 1449, Springer, Berlin, Heilderberg, New York, 1990, Corollary III.2.12, p. 104) which shows that there is no interpolating minimal projection from C[a,b] onto the space of polynomials of degree ≤ n, n ≥ 2). Moreover, this minimal projection is unique (Theorem 9). In particular, Theorem 6 holds for polynomial spaces, generalizing a result of Prophet [(J. Approx. Theory 85 (1996) 27), Theorem 2.1].

SOME CONDITIONS FOR COMPACTNESS IN BC(Q) AND THEIR APPLICATION TO BOUNDARY VALUE PROBLEMS

In this note we present some simple corollaries of an old result of Bartle characterizing the compactness in the space BC(Q) of continuous and bounded real functions defined on a topological space Q. We formulate remarks to obtain conditions so that a subset F of BC(Q) results relatively compact. Some examples and counterexamples shedding light on the structure of compact subsets in BC(Q) are constructed. The results seem to be very useful in the study of Boundary Value Problems on unbounded intervals. AMS Subject Classification: 46E15, 34B15,54C35

Kolmogrov's type criteria for spaces of compact operators

Abstract The aim of this paper is to prove various Kolmogorovs type criteria for spaces of compact operators. We also present the results concerning strongly unique best approximation. In particular we generalize some well known theorems from the theory of minimal projections. As an application, we characterize SUBA projections onto hyperplanes in l∞n and estimate the strong strong unicity constant in this case.