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W1970402582 endingPage "273" @default.
W1970402582 startingPage "261" @default.
W1970402582 abstract "In this paper we study the commutant of an analytic Toeplitz operator. For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi upper H Superscript normal infinity> <mml:semantics> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mspace width=thickmathspace /> <mml:mspace width=thickmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant=normal>∞</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>phi ;;{H^infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi equals chi upper F> <mml:semantics> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>χ</mml:mi> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>phi = chi F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be its inner-outer factorization. Our main result is that if there exists <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda epsilon upper C> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mspace width=thickmathspace /> <mml:mi>ϵ</mml:mi> <mml:mspace width=thickmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>C</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda ;epsilon ;{text {C}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <italic>X</italic> factors as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi equals chi 1 chi 2 midline-horizontal-ellipsis chi Subscript n> <mml:semantics> <mml:mrow> <mml:mi>χ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>χ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo>⋯</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>χ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>chi = {chi _1}{chi _2} cdots {chi _n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, each <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi Subscript i> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>χ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{chi _i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an inner function, and if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F minus lamda> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>−</mml:mo> <mml:mi>λ</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>F - lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is divisible by each <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi Subscript i> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>χ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{chi _i}</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=left-brace upper T Subscript phi Baseline right-brace prime equals left-brace upper T Subscript chi Baseline right-brace prime intersection left-brace upper T Subscript upper F Baseline right-brace prime> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>χ</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>∩</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>F</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{ {T_phi }} ’ = { {T_chi }} ’ cap { {T_F}} ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The key step in the proof is Lemma 2, which is a curious result about nilpotent operators. One corollary of our main result is that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi left-parenthesis z right-parenthesis equals z Superscript n Baseline comma n greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>χ</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>chi (z) = {z^n},n geq 1</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=left-brace upper T Subscript phi Baseline right-brace prime equals left-brace upper T Subscript chi Baseline right-brace prime intersection left-brace upper T Subscript upper F Baseline right-brace prime> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>χ</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>∩</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>F</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{ {T_phi }} ’ = { {T_chi }} ’ cap { {T_F}} ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, another is that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi epsilon upper H Superscript normal infinity> <mml:semantics> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mspace width=thickmathspace /> <mml:mi>ϵ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant=normal>∞</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>phi ;epsilon {H^infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is univalent then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace upper T Subscript phi Baseline right-brace prime equals left-brace upper T Subscript z Baseline right-brace prime> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>z</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{ {T_phi }} ’ = { {T_z}} ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We are also able to prove that if the inner factor of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding=application/x-tex>phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi left-parenthesis z right-parenthesis equals z Superscript n Baseline comma n greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>χ</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>chi (z) = {z^n},n geq 1</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=left-brace upper T Subscript phi Baseline right-brace prime equals left-brace upper T Subscript z Sub Superscript s Subscript Baseline right-brace prime> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{ {T_phi }} ’ = { {T_{{z^s}}}} ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <italic>s</italic> is a positive integer maximal with respect to the property that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=z Superscript n> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{z^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F left-parenthesis z right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are both functions of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=z Superscript s> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{z^s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We conclude by raising six questions." @default.
W1970402582 created "2016-06-24" @default.
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W1970402582 date "1973-01-01" @default.
W1970402582 modified "2023-09-23" @default.
W1970402582 title "The commutant of analytic Toeplitz operators" @default.
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