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- W1569593075 abstract "We show that a composition operator on the Smirnov class <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Superscript plus> <mml:semantics> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>N^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact if and only if it is compact on some (equivalently: <italic>every</italic>) Hardy space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript p> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>H^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than p greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>0>p>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Along the way we show that for composition operators on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Superscript plus> <mml:semantics> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>N^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> both the formally weaker notion of <italic>boundedness,</italic> and a formally stronger notion we call <italic>metric compactness,</italic> are equivalent to compactness." @default.
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- W1569593075 date "1999-12-08" @default.
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- W1569593075 title "Compact composition operators on the Smirnov class" @default.
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- W1569593075 doi "https://doi.org/10.1090/s0002-9939-99-05239-9" @default.
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