Minimal shape-preserving projections in tensor product spaces

JOURNAL OF APPROXIMATION THEORY

卷:162

Article

Lewicki, Grzegorz²  Prophet, Michael¹ 

[1] Univ No Iowa, Dept Math, Cedar Falls, IA 50614 USA

[2] Jagiellonian Univ, Dept Math, PL-30059 Krakow, Poland

关键词: Tensor product;    Minimal projection;    Shape-preserving projection;   

DOI  :  10.1016/j.jat.2009.09.009

来源: Elsevier

【 摘 要 】

Let X denote a (real) Banach space. If P : X -> X is a linear operator and S subset of X such that PS subset of S then we say that S is invariant under P. In the case that P is a projection and S is a cone we say that P is a shape-preserving projection (relative to S) whenever P leaves S invariant. If we assume that cone S has a particular structure then, given a finite-dimensional subspace V subset of X, we can describe, in geometric terms, the set of all shape-preserving projections (relative to S) from X onto V. From here (assuming that such projections exist), we can then look for those shape-preserving projections P : X -> V of the minimal operator norm; that is, we look for minimal shape-preserving projections. If P-i : X-i -> V-i is a minimal shape-preserving projection (relative to S-i) defined on Banach space X-i for i = 1, 2 then it is obvious that P-1 circle times P-2 is a shape-preserving projection (relative to S-1 circle times S-2) on X-1 circle times X-2. But is it true that P-1 circle times P-2 must have minimal norm? In this paper we show that in general this need not be the case (note that this is somewhat unexpected since, in the standard minimal projection setting, the tensor of two minimal projections is always minimal). We also identify a collection of operators in which P-1 circle times P-2 is always a minimal shape-preserving projection (within that collection). This result is then applied to a (well-known) special case to reveal a (non-trivial) situation in which P-1 circle times P-2 is indeed a minimal shape-preserving projection (among all possible shape-preserving projections). (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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