Convergence in trace ideals

Abstract

We give an elementary proof of a theorem of Arazy which presents necessary and sufficient conditions on a symmetric sequence so that the associated symmetrically normed trace ideal has the property that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript n Baseline right-arrow upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_n} \to A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the weak operator topology and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper A Subscript n Baseline double-vertical-bar right-arrow double-vertical-bar upper A double-vertical-bar"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>A</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \| {{A_n}} \right \| \to \left \| A \right \|</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper A Subscript n Baseline minus upper A double-vertical-bar right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \| {{A_n} - A} \right \| \to 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Proceedings of the American Mathematical Society - View - PDF