Minimal shape-preserving projections in tensor product spaces

Abstract

Let X denote a (real) Banach space. If P : X → X is a linear operator and S ⊂ X such that P S ⊂ 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 ⊂ 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 ⊗ P 2 is a shape-preserving projection (relative to S 1 ⊗ S 2 ) on X 1 ⊗ X 2 . But is it true that P 1 ⊗ 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 ⊗ 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 ⊗ P 2 is indeed a minimal shape-preserving projection (among all possible shape-preserving projections).