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- W4376105989 abstract "Abstract The aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if $$Xwidehat{otimes }_pi Y$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>X</mml:mi> <mml:msub> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>π</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> is strongly subdifferentiable and either X or Y has the metric approximation property then every bounded operator from X to $$Y^*$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> is compact. We also prove that $$(ell _p(I)widehat{otimes }_pi ell _q(J))^*$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>π</mml:mi> </mml:msub> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>J</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> has the $$w^*$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>w</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -Kadec-Klee property for every non-empty sets I , J and every $$2<p,q<infty $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , obtaining in particular that the norm of the space $$ell _p(I)widehat{otimes }_pi ell _q(J)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>π</mml:mi> </mml:msub> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>J</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$Xwidehat{otimes }_pi Y$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>X</mml:mi> <mml:msub> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>π</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> is dense but whose complement is dense too." @default.
- W4376105989 created "2023-05-12" @default.
- W4376105989 creator A5068732387 @default.
- W4376105989 date "2023-05-09" @default.
- W4376105989 modified "2023-09-26" @default.
- W4376105989 title "Several Remarks on Norm Attainment in Tensor Product Spaces" @default.
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