Several Remarks on Norm Attainment in Tensor Product Spaces

Abstract

AbstractThe aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if $$X\widehat{\otimes }_\pi Y$$ X ⊗ ^ π Y is strongly subdifferentiable and either X or Y has the metric approximation property then every bounded operator from X to $$Y^*$$ Y ∗ is compact. We also prove that $$(\ell _p(I)\widehat{\otimes }_\pi \ell _q(J))^*$$ ( ℓ p ( I ) ⊗ ^ π ℓ q ( J ) ) ∗ has the $$w^*$$ w ∗ -Kadec-Klee property for every non-empty sets I, J and every $$2<p,q<\infty $$ 2 < p , q < ∞ , obtaining in particular that the norm of the space $$\ell _p(I)\widehat{\otimes }_\pi \ell _q(J)$$ ℓ p ( I ) ⊗ ^ π ℓ q ( J ) is strongly subdifferentiable. This extends several results of Dantas, Kim, Lee and Mazzitelli. We also find examples of spaces X and Y for which the set of norm-attaining tensors in $$X\widehat{\otimes }_\pi Y$$ X ⊗ ^ π Y is dense but whose complement is dense too.