In this paper we shall use realization theory, a favourite technique of Rien Kaashoek, to prove new results about a class of holomorphic functions on an annulus <Equation ID="Equ89"> <EquationSource Format="TEX">$ R_\delta {\mathop {=}\limits ^\textrm{def}}\{z\in \mathbb {C}: \delta<|z|<1\}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>R</mi> <mi>δ</mi> </msub> <mover> <mo>=</mo> <mtext>def</mtext> </mover> <mrow> <mo stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mi>δ</mi> <mo><</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<\delta <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>δ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The class of functions in question arises in the early work of R. G. Douglas and V. I. Paulsen on the rational dilation of a Hilbert space operator T to a normal operator with spectrum in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\partial R_\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <msub> <mi>R</mi> <mi>δ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Their work suggested the following norm <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Vert \cdot \Vert _{\textrm{dp}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mo>·</mo> <mo stretchy="false">‖</mo> </mrow> <mtext>dp</mtext> </msub> </math></EquationSource> </InlineEquation> on the space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{Hol}(R_\delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Hol</mtext> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mi>δ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of holomorphic functions on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R_\delta $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation>, <Equation ID="Equ90"> <EquationSource Format="TEX">$ \Vert \varphi \Vert _{\textrm{dp}} {\mathop {=}\limits ^\textrm{def}} \sup \{ \Vert \varphi (T)\Vert : \Vert T\Vert \le 1, \Vert T^{-1}\Vert \le 1/\delta \ \text {and} \ \sigma (T)\subseteq R_\delta \}. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>φ</mi> <mo stretchy="false">‖</mo> </mrow> <mtext>dp</mtext> </msub> <mover> <mo>=</mo> <mtext>def</mtext> </mover> <mrow> <mo movablelimits="true">sup</mo> <mo stretchy="false">{</mo> <mo stretchy="false">‖</mo> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> <mo>:</mo> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> <mo>,</mo> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>T</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>δ</mi> <mspace width="4pt" /> <mtext>and</mtext> <mspace width="4pt" /> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊆</mo> <msub> <mi>R</mi> <mi>δ</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>By analogy with the classical Schur class of holomorphic functions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {S} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation> with supremum norm at most 1 on the disc <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation>, it is natural to consider the dp-Schur class <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {S}_\textrm{dp}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">S</mi> <mtext>dp</mtext> </msub> </math></EquationSource> </InlineEquation> of holomorphic functions of dp-norm at most 1 on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$R_\delta $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation>. Our central result is a Pick interpolation theorem for functions in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {S}_\textrm{dp}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">S</mi> <mtext>dp</mtext> </msub> </math></EquationSource> </InlineEquation> that is analogous to Abrahamse’s Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\lambda =(\lambda _1,\dots ,\lambda _n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of distinct interpolation nodes in <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$R_\delta $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation>, we introduce a special set <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {G}_\textrm{dp}(\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">G</mi> <mtext>dp</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of positive definite <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> matrices, which we call DP Szegő kernels. The DP Pick problem <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\lambda _j \mapsto z_j, j=1,\dots ,n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>j</mi> </msub> <mo>↦</mo> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, is shown to be solvable if and only if, <Equation ID="Equ91"> <EquationSource Format="TEX">$ {[}(1-{\overline{z}}_i z_j)g_{ij}] \ge 0 \; \text { for all}\; g \in \mathcal {G}_{\textrm{dp}} (\lambda ). $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo stretchy="false">[</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msub> <mover> <mi>z</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>g</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mrow> <mo stretchy="false">]</mo> <mo>≥</mo> <mn>0</mn> <mspace width="0.277778em" /> <mspace width="0.333333em" /> <mtext>for all</mtext> <mspace width="0.277778em" /> <mi>g</mi> <mo>∈</mo> </mrow> <msub> <mi mathvariant="script">G</mi> <mtext>dp</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>We prove further that a solvable DP Pick problem has a solution which is a rational function with a finite-dimensional model, an intriguing result which opens up the possibility of a theory of extremal functions from <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\mathcal {S}_\textrm{dp}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">S</mi> <mtext>dp</mtext> </msub> </math></EquationSource> </InlineEquation> analogous to the theory of finite Blaschke products.

Function theory on the annulus in the dp-norm

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2025年11月3日

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Jim Agler,
Zinaida A. Lykova,
N. J. Young

摘要

In this paper we shall use realization theory, a favourite technique of Rien Kaashoek, to prove new results about a class of holomorphic functions on an annulus $ R_\delta {\mathop {=}\limits ^\textrm{def}}\{z\in \mathbb {C}: \delta<|z|<1\}, $ $R_{δ} \overset{def}{=} {z \in C : δ < | z | < 1},$ where $0<\delta <1$ $0 < δ < 1$ . The class of functions in question arises in the early work of R. G. Douglas and V. I. Paulsen on the rational dilation of a Hilbert space operator T to a normal operator with spectrum in $\partial R_\delta $ $\partial R_{δ}$ . Their work suggested the following norm $\Vert \cdot \Vert _{\textrm{dp}}$ ${‖ \cdot ‖}_{dp}$ on the space $\textrm{Hol}(R_\delta )$ $Hol (R_{δ})$ of holomorphic functions on $R_\delta $ $R_{δ}$ , $ \Vert \varphi \Vert _{\textrm{dp}} {\mathop {=}\limits ^\textrm{def}} \sup \{ \Vert \varphi (T)\Vert : \Vert T\Vert \le 1, \Vert T^{-1}\Vert \le 1/\delta \ \text {and} \ \sigma (T)\subseteq R_\delta \}. $ ${‖ φ ‖}_{dp} \overset{def}{=} sup {‖ φ (T) ‖ : ‖ T ‖ \leq 1, ‖ T^{- 1} ‖ \leq 1 / δ and σ (T) \subseteq R_{δ}} .$ By analogy with the classical Schur class of holomorphic functions $\mathcal {S} $ $S$ with supremum norm at most 1 on the disc $\mathbb {D}$ $D$ , it is natural to consider the dp-Schur class $\mathcal {S}_\textrm{dp}$ $S_{dp}$ of holomorphic functions of dp-norm at most 1 on $R_\delta $ $R_{δ}$ . Our central result is a Pick interpolation theorem for functions in $\mathcal {S}_\textrm{dp}$ $S_{dp}$ that is analogous to Abrahamse’s Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple $\lambda =(\lambda _1,\dots ,\lambda _n)$ $λ = (λ_{1}, \dots, λ_{n})$ of distinct interpolation nodes in $R_\delta $ $R_{δ}$ , we introduce a special set $\mathcal {G}_\textrm{dp}(\lambda )$ $G_{dp} (λ)$ of positive definite $n\times n$ $n \times n$ matrices, which we call DP Szegő kernels. The DP Pick problem $\lambda _j \mapsto z_j, j=1,\dots ,n$ $λ_{j} \mapsto z_{j}, j = 1, \dots, n$ , is shown to be solvable if and only if, $ {[}(1-{\overline{z}}_i z_j)g_{ij}] \ge 0 \; \text { for all}\; g \in \mathcal {G}_{\textrm{dp}} (\lambda ). $ $[(1 - {\bar{z}}_{i} z_{j}) g_{ij}] \geq 0 for all g \in G_{dp} (λ) .$ We prove further that a solvable DP Pick problem has a solution which is a rational function with a finite-dimensional model, an intriguing result which opens up the possibility of a theory of extremal functions from $\mathcal {S}_\textrm{dp}$ $S_{dp}$ analogous to the theory of finite Blaschke products.