Composition operators isolated in the uniform operator topology

Abstract

It is is shown that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an analytic map of the disc <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue z EndAbsoluteValue greater-than 1"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>z</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\left | z \right | &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into itself such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has radial limits of modulus 1 on a set of positive measure, then for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-slanted-equals p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leqslant p &gt; \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the corresponding composition operator on <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:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isolated in the topological space of composition operators on <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:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (with the uniform operator topology).

Locations

• Proceedings of the American Mathematical Society - View - PDF