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<p>We consider de Branges–Rovnyak spaces of a considerably large class of reproducing kernel Hilbert spaces and find a characterization for them to be complete Nevanlinna–Pick spaces. This extends earlier characterizations obtained for the Hardy space over the unit disc (Chu in J Funct Anal 279:1–15, 2020) as well as for the Drury–Arveson space over the unit ball (Sautel in PhD diss. University of Tennessee, 2022. <a href="https://trace.tennessee.edu/utk_graddiss/7144/">https://trace.tennessee.edu/utk_graddiss/7144/</a>). Our characterization takes a complete form for the particular cases of the Hardy space over the polydisc and the Bergman space over the disc. We show that a non-trivial de Branges–Rovnyak space, associated to a contractive multiplier, of the Hardy space over the bidisc or the Bergman space over the unit disc is a complete Nevanlinna–Pick space if and only if it is isometrically isomorphic to the Hardy space over the unit disc as reproducing kernel Hilbert spaces. On the contrary, it is shown that non-trivial de Branges–Rovnyak spaces of the Hardy space over the <i>n</i>-disc with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2024_1881_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> are never complete Nevanlinna–Pick spaces.</p>

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De Branges–Rovnyak Spaces Which are Complete Nevanlinna–Pick Spaces

  • 期刊论文

  • 发表日期：

    2025年1月4日

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  • Hamidul Ahmed,
  • B. Krishna Das,
  • Samir Panja

摘要

We consider de Branges–Rovnyak spaces of a considerably large class of reproducing kernel Hilbert spaces and find a characterization for them to be complete Nevanlinna–Pick spaces. This extends earlier characterizations obtained for the Hardy space over the unit disc (Chu in J Funct Anal 279:1–15, 2020) as well as for the Drury–Arveson space over the unit ball (Sautel in PhD diss. University of Tennessee, 2022. https://trace.tennessee.edu/utk_graddiss/7144/). Our characterization takes a complete form for the particular cases of the Hardy space over the polydisc and the Bergman space over the disc. We show that a non-trivial de Branges–Rovnyak space, associated to a contractive multiplier, of the Hardy space over the bidisc or the Bergman space over the unit disc is a complete Nevanlinna–Pick space if and only if it is isometrically isomorphic to the Hardy space over the unit disc as reproducing kernel Hilbert spaces. On the contrary, it is shown that non-trivial de Branges–Rovnyak spaces of the Hardy space over the n-disc with \(n\ge 3\) n ≥ 3 are never complete Nevanlinna–Pick spaces.