Faculty Publications


Minimal shape-preserving projections in tensor product spaces

Document Type



Minimal projection, Shape-preserving projection, Tensor product

Journal/Book/Conference Title

Journal of Approximation Theory





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Let X denote a (real) Banach space. If P:X→X is a linear operator and ScyrillicX such that PScyrillicS 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 VcyrillicX, 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 Pi:Xi→Viis a minimal shape-preserving projection (relative to Si) defined on Banach space Xi for i=1,2 then it is obvious that P1×P2is a shape-preserving projection (relative to S1×S2) on X1×X2. But is it true that P1×P2 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 P1×P2 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 P1×P2 is indeed a minimal shape-preserving projection (among all possible shape-preserving projections). © 2009 Elsevier Inc.

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