Invariant subspaces in Banach spaces of analytic functions

Abstract

We study the invariant subspace structure of the operator of multiplication by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z"> <mml:semantics> <mml:mi>z</mml:mi> <mml:annotation encoding="application/x-tex">z</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>z</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the Cowen-Douglas class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B 1 left-parenthesis normal upper Omega overbar right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {B}_1}(\overline \Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We say that an invariant subspace <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c o d script upper M equals 1"> <mml:semantics> <mml:mrow> <mml:mi>cod</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {cod} \mathcal {M} = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z script upper M"> <mml:semantics> <mml:mrow> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">z\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has codimension one in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We give various conditions on invariant subspaces which imply that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c o d script upper M equals 1"> <mml:semantics> <mml:mrow> <mml:mi>cod</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {cod} \mathcal {M} = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, we give a necessary and sufficient condition on two invariant subspaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c o d script upper M equals c o d script upper N equals 1"> <mml:semantics> <mml:mrow> <mml:mi>cod</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>cod</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {cod} \mathcal {M} = \operatorname {cod} \mathcal {N} = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that their span again satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c o d left-parenthesis script upper M logical-or script upper N right-parenthesis equals 1"> <mml:semantics> <mml:mrow> <mml:mi>cod</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo>∨</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {cod} (\mathcal {M} \vee \mathcal {N}) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This result will be used to show that any invariant subspace of the Bergman space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript a Superscript p Baseline comma p greater-than-or-slanted-equals 1"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>a</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>p</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">L_a^p,\,p \geqslant 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is generated by functions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript a Superscript 2 p"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">L_a^{2p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, must satisfy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c o d script upper M equals 1"> <mml:semantics> <mml:mrow> <mml:mi>cod</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {cod} \mathcal {M} = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For an invariant subspace <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we then consider the operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S equals upper M Subscript z Superscript asterisk Baseline vertical-bar script upper M Superscript up-tack"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow cla