Reflexivity and Weak Sequential Completeness in Operator Ideals and Polynomial Ideals

Abstract Let E and F be Banach spaces. We establish conditions under which weak sequential completeness is equivalent to reflexivity in the space $$\mathcal{I}(E;F)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>;</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of linear operators from E to F belonging to a given operator ideal $$\mathcal I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>I</mml:mi> </mml:math> . Additionally, we investigate similar conditions for the space $$\mathcal{Q}(^nE;F)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:msup> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> </mml:msup> <mml:mi>E</mml:mi> <mml:mo>;</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of n -homogeneous polynomials from E to F belonging to a given polynomial ideal $$\mathcal Q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> . Illustrative examples are provided.