Lesław Skrzypek

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

Minimal projections in spaces of functions of N variables

We will construct a minimal and co-minimal projection from Lp([0, 1]n) onto Lp([0, 1]n1) + ... + Lp([0, 1]nk), where n = n1 + ... + nk. (see Theorem 2.9). This is a generalization of a result of Cheney, Halton and Light from (Approximation Theory in Tensor Product Spaces, Lecture Notes in Mathematics, Springer, Berlin, 1985; Math. Proc. Cambridge Philos. Soc. 97 (1985) 127; Math. Z. 191 (1986) 633) where they proved the minimality in the case n = 2. We provide also some further generalizations (see Theorems 2.10 and 2.11 (Orlicz spaces) and Theorem 2.8). Also a discrete case (Theorem 2.2) is considered. Our approach differs from methods used in [8,13,20].

Norming points and unique minimality of orthogonal projections

We study the norming points and norming functionals of symmetric operators on Lp spaces for p=2m or p=2m/(2m−1). We prove some general result relating uniqueness of minimal projection to the set of norming functionals. As a main application, we obtain that the Fourier projection onto span [1,sin⁡x,cos⁡x] is a unique minimal projection in Lp.

On the

The purpose of this paper is to find the exact norm of the Rademacher projection onto {r 1 , r 2 , r 3 }. Namely, we prove formula math. The same techniques also give the relative projection constant of ker{1, ..., 1} in l n p , that is, formula math. for n = 2,3,4. We discuss the relation of the above inequalities to the famous Khintchine and Clarkson inequalities. We conclude the paper by stating some conjectures that involve the geometry of the unit ball of l n p .

L_p

The purpose of this paper is to find the relative projection constant of ker { 1 } = { x : ? i = 1 n x i = 0 } in ? p n for an arbitrary n (see Theorem 2.28). This extends a result of L. Skrzypek obtained for n = 3 , 4 . We also improve on the formula for the relative projection constant onto hyperplanes in L p 0 , 1 obtained by C. Franchetti (see Theorem 2.21) and reprove a result of S. Rolewicz (see Corollary 2.11).

norm of the Rademacher projection and related inequalities

In this paper, we explore the relation between the minimal and the orthogonal projections onto hyperplanes in \(\ell _1^n\) and \(\ell _\infty ^n.\)

Minimal projections onto hyperplanes in ℓ p n

We construct the Chalmers–Metcalf operator for minimal projections onto hyperplanes in l ∞ n and l 1 n and prove it is uniquely determined. We show how we can use Chalmers–Metcalf operator to obtain uniqueness of minimal projections. The main advantage of our approach is that it is purely algebraical and does not require consideration of the min–max problems.