Jointly cyclic polynomials and maximal domains

For a (not necessarily locally convex) topological vector space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of holomorphic functions in one complex variable, we show that the shift invariant subspace generated by a set of polynomials is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if their common vanishing set contains no point at which the evaluation functional is continuous. For two variables, we show that this problem can be reduced to determining the cyclicity of a single polynomial and obtain partial results for more than two variables. We proceed to examine the maximal domain, i.e., the set of all points for which the evaluation functional is continuous. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is metrizable, we show that the maximal domain must be an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript sigma"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi> σ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> set, and then construct Hilbert function spaces on the unit disk whose maximal domain is the disk plus an arbitrary subset of the boundary that is both <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript sigma"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi> σ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_\sigma</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 G Subscript delta"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mi> δ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">G_\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .