Title: A reflexivity theorem for weakly closed subspaces of operators
Abstract: It was proved in [<bold>4</bold>] that the ultraweakly closed algebras generated by certain contractions on Hilbert space have a remarkable property. This property, in conjunction with the fact that these algebras are isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, was used in [<bold>3</bold>] to show that such ultraweakly closed algebras are reflexive. In the present paper we prove an analogous result that does not require isomorphism with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and applies even to linear spaces of operators. Our result contains the reflexivity theorems of [<bold>3</bold>,<bold>2</bold> and <bold>9</bold>] as particular cases.