Minimal shape-preserving projections in tensor product spaces

Let XX denote a (real) Banach space. If P:X→XP:X→X is a linear operator and S⊂XS⊂X such that PS⊂SPS⊂S then we say that SS is invariant under PP. In the case that PP is a projection and SS is a cone we say that PP is a shape-preserving projection   (relative to SS) whenever PP leaves SS invariant. If we assume that cone SS has a particular structure then, given a finite-dimensional subspace V⊂XV⊂X, we can describe, in geometric terms, the set of all shape-preserving projections (relative to SS) from XX onto VV. From here (assuming that such projections exist), we can then look for those shape-preserving projections P:X→VP:X→V of the minimal operator norm; that is, we look for minimal shape-preserving projections.If Pi:Xi→ViPi:Xi→Vi is a minimal shape-preserving projection (relative to SiSi) defined on Banach space XiXi for i=1,2i=1,2 then it is obvious that P1⊗P2P1⊗P2 is a shape-preserving projection (relative to S1⊗S2S1⊗S2) on X1⊗X2X1⊗X2. But is it true that P1⊗P2P1⊗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⊗P2P1⊗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⊗P2P1⊗P2 is indeed a minimal shape-preserving projection (among all possible shape-preserving projections).