Several Remarks on Norm Attainment in Tensor Product Spaces

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