Codimension-one minimal projections onto Haar subspaces

Abstract

Let H n be an n-dimensional Haar subspace of X ¼ C R ½a; b and let H nÀ1 be a Haar subspace of H n of dimension n À 1: In this note we show (Theorem 6) that if the norm of a minimal projection from H n onto H nÀ1 is greater than 1, then this projection is an interpolating projection. This is a surprising result in comparison with Cheney and Morris () which shows that there is no interpolating minimal projection from C½a; b onto the space of polynomials of degree pn; ðnX2Þ: Moreover, this minimal projection is unique (Theorem 9). In particular, Theorem 6 holds for polynomial spaces, generalizing a result of Prophet [(J.

Codimension-one minimal extensions onto Haar subspaces

“…Namely, a class of operators A satisfying desired conditions, significantly extends the case A = I d H n−1 considered in [19]. Moreover, we justify Remark 9 by presenting operators A, for which finding an isomorphismà such that ∥Ã∥ < λ A is relatively easy.…”

“…In [19] the uniqueness of P 0 is obtained as a consequence of the fact that minimal projection P 0 ∈ P(H n , H n−1 ) such that ∥P 0 ∥ > 1 is an interpolating operator.…”

“…Since I d H n -interpolating extensions are classical interpolating projections (see (8) and (11)), the result of this note is a generalization of [19], where the case A = I d H n−1 was considered.…”

“…The aim of this paper is to generalize the results from [19] to the case of extensions of operators. We show that for A ∈ L(H n−1 ) satisfying λ A (H n−1 , H n ) > ∥A∥, the minimal extension of A in the set P A (H n , H n−1 ) is strongly unique.…”

On the non-uniqueness of minimal projection in Lp spaces

“…If U is a finite-dimensional or finite-codimensional space then a minimal projection of X onto U always exists [11]. In this note we are concerned with its uniqueness (see [4,[7][8][9] for particular examples of unique minimal projections).…”