April 02, 2024
Using factorisation and Arov-Krein inequality results, we derive important inequalities (in terms of \(S\)-nodes) in interpolation problems.
Faculty of Mathematics, University of Vienna,
Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
E-mail: oleksandr.sakhnovych@univie.ac.at
To the memory of V.E. Katsnelson
MSC(2020): 30E05, 47A48, 47A56, 47A68, 30J99.
Keywords: Interpolation problem, \(S\)-node, factorisation of spectral matrix function, outer matrix function, entropy inequality.
Let us consider some finite or infinite-dimensional Hilbert space \({\mathcal{H}}\) and operators \(A,S\in {\mathcal{B}}({\mathcal{H}})\), \(\Pi \in {\mathcal{B}}({\mathbb{C}}^{2p},{\mathcal{H}})\), which satisfy the operator identity \[\begin{align} & \label{I1} AS-SA^*=\mathrm{i}\Pi J \Pi^*, \quad J:=\begin{bmatrix} 0 & I_p \\ I_p & 0 \end{bmatrix}, \end{align}\tag{1}\] where \({\mathcal{B}}({\mathcal{H}}_1,{\mathcal{H}}_2)\) stands for the set of bounded linear operators acting from the Hilbert space \({\mathcal{H}}_1\) into Hilbert space \({\mathcal{H}}_2\), \({\mathcal{B}}({\mathcal{H}}):={\mathcal{B}}({\mathcal{H}},{\mathcal{H}})\), \({\mathbb{C}}\) is (as usually) the complex plane, the symbol \({\mathbb{C}}^{n\times p}\) denotes the set of \(n\times p\) matrices with complex-valued entries (\({\mathbb{C}}^n={\mathbb{C}}^{n\times 1}\)), \(\mathrm{i}\) stands for the imaginary unit (\(\mathrm{i}^2=-1\)) and \(I_p\) is the \(p\times p\) identity matrix. The operators \(A^*\) and \(\Pi^*\) in 1 are adjoint to \(A\) and \(\Pi\), respectively, and \(S\) is a self-adjoint operator (\(S=S^*\)).
The triple \(\{A,S,\Pi\}\) with the mentioned above properties is called a self-adjoint \(S\)-node \((\)see [1]–[3] and the references therein\()\) or simply \(S\)-node because the \(S\)-nodes are always self-adjoint in this paper.
Using \(S\)-nodes, a wide class of interpolation problems is solved in a general way [3]. For this purpose, \(2p\times 2p\) matrix valued functions (so called frames) \({\mathfrak A}(S,z)\) with the \(p\times p\) blocks \({\mathfrak A}_{ij}(S,z)={\mathfrak A}_{ij}(z)\) are introduced in [3] (see also the references therein): \[\begin{align} & \label{I2} {\mathfrak A}(S,z)=\{{\mathfrak A}_{ij}(z)\}_{i,j=1}^2=w_A(1/\overline{z})^*=I_{2p}-\mathrm{i}z\Pi^*(I-zA^*)^{-1}S^{-1}\Pi J, \end{align}\tag{2}\] where \(I\) is the identity operator and \[\begin{align} & \label{I3} w_A(\lambda)= I_{2p}-\mathrm{i}J\Pi^*S^{-1}\big(A-\lambda I\big)^{-1}\Pi \end{align}\tag{3}\] is the transfer matrix function (matrix valued function) in Lev Sakhnovich form [2], [3]. We partition the operator \(\Pi\) into two blocks \[\begin{align} & \label{I4} \Pi = \begin{bmatrix} \Phi_1 & \Phi_2\end{bmatrix} \quad \big(\Phi_k\in {\mathcal{B}}({\mathbb{C}}^{p},{\mathcal{H}})\big), \end{align}\tag{4}\] which generates the partition of \({\mathfrak A}(S,z)\) into the blocks \({\mathfrak A}_{ij}(z)\).
It is assumed in important interpolation problems (and will be usually assumed here) that \[\begin{align} & \label{I5} S\geq \varepsilon I \quad (\varepsilon>0), \quad {\rm Ker}\, \Phi_2=0, \end{align}\tag{5}\] where \(S\geq \varepsilon I\) may be expressed in terms of the scalar products \((\cdot , \cdot )\) as \((Sf,f)\geq \varepsilon(f,f)\) \((f \in{\mathcal{H}})\) and \({\rm Ker}\) is the kernel (nullspace) of the corresponding operator. (The strict operator inequality \(S_1>S_2\) means that \(\big((S_1-S_2)f,f\big)>0\) for \(f\not=0\).) In view of 5 , \(S^{-1}\) exists and \(S^{-1}\in {\mathcal{B}}({\mathcal{H}})\). Therefore, \({\mathfrak A}(S,z)\) is well defined in the points of invertibility of \(I-zA^*\). We will assume that the operators \(I-zA^*\) have bounded inverse operators \((\)bounded inverses\()\) for \(z\in {\mathbb{C}}_+\), excluding, possibly, some isolated points, which are the poles of the matrix function \({\mathfrak A}(S,z)\). Here, \({\mathbb{C}}_+\) denotes open upper half-plane \(\Im(z)>0\), where \(\Im(z)\) is the imaginary part of \(z\).
It easily follows from 5 and identity 1 (see also [2] or [1]) that \[\begin{align} & \label{I6} {\mathfrak A}(S,z)J {\mathfrak A}(S,\overline{\lambda})^*=J-\mathrm{i}(z-\lambda)\Pi^*(I-zA^*)^{-1}S^{-1}(I-\lambda A)^{-1}\Pi. \end{align}\tag{6}\] It also follows from 5 that \(S^{-1}>0\). Hence, 6 yields that \(\break {\mathfrak A}(S,z)J{\mathfrak A}(S,z)^*\geq J\) for \(\Im(z)>0\) , that is, for \(z\in {\mathbb{C}}_+\). Equivalently (see, e.g., [1]) we have the inequality \[\begin{align} & \label{I7} {\mathfrak A}(S,z)^*J{\mathfrak A}(S,z)\geq J \qquad {\mathrm{for}} \qquad z\in {\mathbb{C}}_+. \end{align}\tag{7}\] Moreover, for the lower right block of \({\mathfrak A}(S,z)^*J{\mathfrak A}(S,z)\), relations 4 –6 yield \[\begin{align} &\nonumber {\mathfrak A}_{21}(z){\mathfrak A}_{22}(z)^*+{\mathfrak A}_{22}(z){\mathfrak A}_{21}(z)^* \\ & \label{I8} = \mathrm{i}(\overline{z}-z)\Phi_2^*(I-zA^*)^{-1}S^{-1}(I-\overline{z} A)^{-1}\Phi_2 >0 \quad \big(z\in {\mathbb{C}}_+\big). \end{align}\tag{8}\] An important equality \[\begin{align} & \label{I843} {\mathfrak A}(S,z)J{\mathfrak A}(S,\overline{z}))^*= J \end{align}\tag{9}\] follows from 5 as well. We will need the following notation.
****Notation** 1**. A pair \(\{R(z), \, Q(z)\}\) is called nonsingular, with property-\(J\) if \(R(z)\) and \(Q(z)\) are meromorphic \(p\times p\) matrix functions in \({\mathbb{C}}_+\) satisfying relations \[\begin{align} & \label{I9} R(z)^*R(z)+Q(z)^*Q(z)>0, \quad \begin{bmatrix}R(z)^* & Q(z)^* \end{bmatrix}J \begin{bmatrix}R(z) \\ Q(z) \end{bmatrix}\geq 0 \end{align}\qquad{(1)}\] \((\)excluding, possibly, some isolated points \(z\in {\mathbb{C}}_+)\).
It follows from 8 and ?? (see, e.g., [1]) that \[\begin{align} & \label{I10} \det\bigl( {\mathfrak A}_{21}(z ) R(z)+{\mathfrak A}_{22}(z)Q(z)\bigr)\not=0. \end{align}\tag{10}\] Thus, the linear-fractional transformations \[\label{I11} \varphi (z) = \mathrm{i}\bigl( {\mathfrak A}_{11}(z)R(z)+ {\mathfrak A}_{12}(z)Q(z)\bigr) \bigl( {\mathfrak A}_{21}(z ) R(z)+{\mathfrak A}_{22}(z)Q(z)\bigr)^{-1}\tag{11}\] are well defined in the points \(z\in{\mathbb{C}}_+\) where \(I-zA^*\) is invertible and ?? holds.
****Notation** 2**. The set of matrix functions \(\varphi(z)\) of the form 11 , which correspond to some fixed \(S\)-node and various nonsingular pairs with proper-ty-\(J\) is denoted by \({\mathcal{N}}({\mathfrak A}(S))\).
According to 7 , ?? and 11 , we have \[\begin{align} & \label{I12} \mathrm{i}(\varphi(z)^*-\varphi(z))\geq 0 \quad (z\in {\mathbb{C}}_+). \end{align}\tag{12}\] This implies that the matrix functions \(\varphi(z)\) do not have singularities in \({\mathbb{C}}_+\) (even if \({\mathfrak A}(S,z)\) has poles), belong to Herglotz class and admit Herglotz representation: \[\begin{align} & \tag{13} \varphi(z )=\gamma z +\theta+\int_{-\infty}^{\infty}\frac{1+t z }{(t-z )(1+t^2)}d\mu(t) \quad (\gamma\geq 0, \quad \theta=\theta^*), \\ & \tag{14} \int_{-\infty}^{\infty}{(1+t^2)^{-1}}{d\mu(t)}<\infty, \end{align}\] where \(\mu(t)\) is a nondecreasing \(p\times p\) matrix function (and \(d\mu\) is often called matrix measure). The solutions of interpolation problems for the structured operators \(S\) are presented in [3] as the triples \(\{\gamma,\theta,\mu\}\). They are described using linear-fractional transformations 11 . The expressions \[\begin{align} & \label{I15} \rho(z, \lambda):=\mathrm{i}(\lambda-z)\Phi_2^*(I-zA^*)^{-1}S^{-1}(I-\lambda A)^{-1}\Phi_2 \end{align}\tag{15}\] and \(\rho(z, \overline{z})\) are important characteristics of the structured operators \(S\). Therefore, the interrelations between \(\rho(z, \overline{z})\) and \(\mu\), which we study in this paper, are of essential interest in interpolation and asymptotic analysis.
In Section 2, the factorisation \(\mu^{\prime}(t)=G_{\mu}(t)^*G_{\mu}(t)\) is considered. Some results and references from the important survey [4] by V.E. Katsnelson and B. Kirstein were used in that section. We note that our Theorem 8 on the interrelations between \(\rho(z, \overline{z})\) and \(\mu\) is expressed in terms of \(G_{\mu}(z)\) (see some examples in [5], [6]). The interesting note [7] by D.Z. Arov and M.G. Krein is applied in Section 3 in order to obtain this theorem. In the spirit of [7], the inequality ?? in Theorem 8 may be called the entropy inequality. Finally, in Section 4 we formulate and prove a generalisation (without various restrictions from the earlier interpolation theorem [8]) of [5], which gives sufficient conditions for the relations 38 to hold. Relations 38 are the main requirements in Theorem 8. We note that the proof of [5] (generalised here) was not published in [5] or later.
Notations. The majority of the notations have been explained above. In addition, the notation \({\mathbb{D}}\) stands for the unit disk \(|\zeta|<1\), the notation \({\rm tr}(Z)\) stands for the trace of \(Z\), and \(\Re(Z)\) denotes the real part (of either a scalar or a square matrix \(Z\)).
Factorisation of positive semi-definite integrable matrix functions is one of the classical domains connected with the names of A. Beurling, N. Wiener, P.R. Masani, H. Helson, D. Lowdenslager, M.G. Krein, D. Sarason and many others (see, e.g., [9] and the useful bibliography in [4]). The main results deal with the functions and matrix functions on the unit disk. The results on the real axis follow from the classical conformal mappings between \({\mathbb{C}}_+\) and the unit disk \({\mathbb{D}}\). Recall that the functions \[\begin{align} & \label{A1} z=(\overline{z_0}\zeta -z_0 )(\zeta -1)^{-1} \qquad (z_0\in {\mathbb{C}}_+) \end{align}\tag{16}\] map the unit disk onto \({\mathbb{C}}_+\). In spite of the simplicity of these mappings, it would be convenient to reformulate the factorisation results in terms of the real axis (so far, we know the book [10], where the scalar case is treated, and some constructive results in [11] for the matrix function case). Meanwhile, following [6] we introduce the class \(\widehat H\) of matrix functions on \({\mathbb{C}}_+\). First recall that an outer (or maximal in the terminology of [12]) function \(h(\zeta)\) is an analytic in \({\mathbb{D}}\) function, which admits representation \[\begin{align} & \label{A233} h(\zeta)=\alpha\exp\left\{\frac{1}{2\pi}\int_{0}^{2\pi}\ln\big(\omega(\vartheta)\big)\frac{\mathrm{e}^{\mathrm{i}\vartheta}+ \zeta}{\mathrm{e}^{\mathrm{i}\vartheta}- \zeta} d\vartheta\right\}, \end{align}\tag{17}\] where \(|\alpha|=1\) and \(\ln\big(\omega(\vartheta)\big)\) is integrable on \([0,2\pi]\). Clearly, the product of outer matrix functions is outer as well. Representation 17 yields that \(\omega(\vartheta)=\big|h\big(\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|\). (We note that functions on \({\mathbb{C}}_+\) and \({\mathbb{D}}\) considered in this paper have non-tangential boundary values almost everywhere on \({\mathbb{R}}\) and unit circle \(|\zeta|=1\), respectively.)
****Definition** 3**. The \(p \times p\) matrix function \(G(z)\) belongs to \(\widehat H\) if the entries of \[\begin{align} & \label{A143} \widehat G(\zeta)=G\big((\overline{z_0}\zeta -z_0 )(\zeta -1)^{-1}\big) \end{align}\qquad{(2)}\] belong to the Hardy class \(H^2({\mathbb{D}})\) and \(\det \left(\widehat G(\zeta)\right)\) is an outer function.
****Remark** 4**. We note that the accent “widehat" denotes \((\)as in ?? \()\) the transfer from the matrix function depending on \(z\in {\mathbb{C}}_+\) to the matrix function depending on \(\zeta\in {\mathbb{D}}\) via the substitution \(z=(\overline{z_0}\zeta -z_0 )(\zeta -1)^{-1}\).
The accent “breve" denotes the inverse mapping: \[\begin{align} & \label{A2-} \breve f(z)=f\big((z-z_0)/(z-\overline{z_0})\big). \end{align}\qquad{(3)}\]
The corresponding limit functions on \({\mathbb{R}}\) and on the unit circle are also denoted by \(\breve f\) and \(\widehat G\), respectively.
****Remark** 5**. In order to invert matrix functions we are interested in, we recall some definitions and properties of Smirnov class \(D\) of functions and of the outer matrix functions, which are used in this paper and in [4], [7] as well. A holomorphic in \({\mathbb{D}}\) function \(f\) belongs to \(D\) if it may be represented as a ratio of a function from \(H^{\infty}({\mathbb{D}})\) and of an outer function from \(H^{\infty}({\mathbb{D}})\) \((\)see, e.g., [13]\()\). In particular, the function \(h(z)\) given in 17 and \(1/h(z)\) belong \(D\). If \(f,g\in D\), than \(fg\in D\) and \((f+g)\in D\). According to another \((\)equivalent\()\) definition of \(D\) from [12] and Polubarinova-Kochina theorem [12], all the functions from the Hardy classes \(H^{\delta}({\mathbb{D}})\) belong to \(D\). The notation \(D^{(p\times \ell)}\) stands for the class of \(p\times \ell\) matrix functions with the entries belonging to \(D\). A matrix function \(f\) is called an outer matrix function if \(f\in D^{p\times p}\) and its determinant is an outer function [13]. From the facts above, it is clear that the inverse to the outer matrix function exists and is again an outer matrix function.
Next, assume that a nondecreasing matrix function \(\mu(t)\) satisfies 14 and Szegő condition \[\begin{align} & \label{A2} \int_{-\infty}^{\infty}(1+t^2)^{-1}\ln\big(\det\mu^{\prime}(t)\big)dt>-\infty , \end{align}\tag{18}\] where \(\mu^{\prime}\) is the derivative of the absolutely continuous part of \(\mu\). Setting \[\begin{align} & \nonumber t=(\overline{z_0}\zeta -z_0 )(\zeta -1)^{-1}=\overline{z_0}+(\overline{z_0}-{z_0})(\zeta-1)^{-1}, \quad \zeta=\mathrm{e}^{\mathrm{i}\theta} \quad( 0\leq \theta <2\pi), \end{align}\] and taking into account that \(\overline{\zeta}(\zeta-1)=-\overline{(\zeta-1)}\) and \(\overline{\zeta}(\overline{z_0}\zeta -z_0 )=-\overline{\big(\overline{z_0}\zeta -z_0 \big)}\) we obtain \[\begin{align} & \label{A3} \frac{dt}{1+t^2}=-\frac{\mathrm{i}(\overline{z_0}-{z_0})\zeta d\theta}{(\zeta-1)^2+(\overline{z_0}\zeta -z_0 )^2}= \frac{\mathrm{i}(\overline{z_0}-{z_0})d\theta}{|\zeta-1|^2+|\overline{z_0}\zeta -z_0 |^2}. \end{align}\tag{19}\] From 14 and 19 , it follows that \[\int_0^{2\pi}\mu^{\prime}\left(\frac{\overline{z_0}\mathrm{e}^{\mathrm{i}\theta} -z_0 }{\mathrm{e}^{\mathrm{i}\theta} -1}\right)d\theta<\infty.\] In other words, \(\mu^{\prime}\left(\frac{\overline{z_0}\mathrm{e}^{\mathrm{i}\theta} -z_0 }{\mathrm{e}^{\mathrm{i}\theta} -1}\right)\) is integrable. Since \(\mu^{\prime}(t)\geq 0\) and \(\mu^{\prime}\left(\frac{\overline{z_0}\mathrm{e}^{\mathrm{i}\theta} -z_0 }{\mathrm{e}^{\mathrm{i}\theta} -1}\right)\) is integrable, the first two conditions of [9] are fulfilled.
From 18 and 19 we derive that \(\int_0^{2\pi}\ln\det\left(\mu^{\prime}\left(\frac{\overline{z_0}\mathrm{e}^{\mathrm{i}\theta} -z_0 }{\mathrm{e}^{\mathrm{i}\theta} -1}\right)\right)d\theta>-\infty\), and, in view of the equality \(\ln(\det (\mu^{\prime}))= {\rm tr}(\ln (\mu^{\prime}))\), the condition (74) of [9] is also fulfilled. Thus (see also [4]), we have the following proposition.
****Proposition** 6**. Let a nondecreasing matrix function \(\mu(t)\) satisfy 14 and Szegő condition 18 . Then, there is a factorisation \[\begin{align} & \label{A4} \mu^{\prime}(t)=G_{\mu}(t)^*G_{\mu}(t), \end{align}\qquad{(4)}\] where \(G_{\mu}(z )\in \widehat H\) and \(G_{\mu}(t)\) is the boundary value function of \(G_{\mu}(z )\). Moreover, \(G_\mu(z)\) may be chosen uniquely up to a constant unitary factor from the left.
In the proposition above, we reversed the order of factors in [9], which does not matter (see, e.g., [9]). The uniqueness of \(G_{\mu}(z)\) (up to a constant unitary factor from the left). follows, for instance, from [4]. Indeed, let \(G_{\mu}\in \widehat H\) and \(F_{\mu}\in \widehat H\) satisfy ?? . Than, \(\widehat G_{\mu}(\zeta) \widehat F_{\mu}(\zeta)^{-1}\) is unitary for \(|\zeta|=1\). According to the “maximum principle of V.I. Smirnov for matrix functions" (see [4]), \(\widehat G_{\mu}(\zeta) \widehat F_{\mu}(\zeta)^{-1}\) belongs also to the Schur class of contractive matrix functions. Therefore, \(\widehat G_{\mu}(\zeta) \widehat F_{\mu}(\zeta)^{-1}\) is an inner matrix function. Clearly, \(\widehat G_{\mu}(\zeta) \widehat F_{\mu}(\zeta)^{-1}\) is invertible and its inverse belongs \(D^{p\times p}\). Now, in view of [4], we see that \(\widehat G_{\mu}(\zeta) \widehat F_{\mu}(\zeta)^{-1}\) is a constant (and so unitary) matrix.
****Remark** 7**. The definition of \(\widehat H\) does not depend on the choice of \(\break z_0\in {\mathbb{C}}_+\) because \(f(\zeta)\in D\) yields \[g(\eta):=f\left(\frac{a\eta-b}{\overline{b}\eta-\overline{a}} \right)\in D \quad {\mathrm{for}} \quad b\not=0, \,\, |a|>|b|, \,\, |\eta|<1.\]
Interesting inequalities (and asymptotic results) based on linear fractional transformations have been published by D.Z. Arov and M.G. Krein in [7] (see also some related proofs in [14]). In order to use them in our work, certain reformulations are necessary.
The \(2p \times 2p\) meromorphic matrix functions \({\mathcal{A}}(\zeta)\) (\(|\zeta|< 1\)), such that \[\begin{align} & \label{B1} {\mathcal{A}}(\zeta)^*J{\mathcal{A}}(\zeta)\geq -j , \quad j:=\begin{bmatrix} I_p & 0 \\ 0 & -I_p \end{bmatrix}, \end{align}\tag{20}\] have been considered in [7]. Note that changes in this section of several notations from [7] were caused by the notations used in the previous sections. For instance, the notation \(A(z)\) have been used in [7] instead of \({\mathcal{A}}(\zeta)\) and \(n\) have been used instead of \(p\).
Let us assume that an \(S\)-node \(\{A,S,\Pi\}\) is given, that is, the identity 1 holds, \(A,\, S \in {\mathcal{B}}({\mathcal{H}})\), and \(S=S^*\). Assume that \(S\) satisfies the first relation in 5 and that the operators \(I-zA^*\) have bounded inverses for \(z\in {\mathbb{C}}_+\) (excluding, possibly, some isolated points, which are the poles of the matrix function \({\mathfrak A}(S,z)\) given by 2 ).
It is easy to see that \(J\) and \(j\) are unitarily similar: \[\begin{align} & \label{c13} J=KjK^*, \quad {\mathrm{where}} \quad K:= \frac{1}{\sqrt{2}}\begin{bmatrix}I_p & -I_p \\ I_p & I_p \end{bmatrix}, \quad K^*=K^{-1}. \end{align}\tag{21}\] Therefore, we also have \(J=KJ(-j)JK^*\) and so 7 implies that the matrix function \[\begin{align} & \label{B3} {\mathcal{A}}(\zeta):=\widehat{\mathfrak A}(S,\zeta)KJ, \end{align}\tag{22}\] where \(\widehat{\mathfrak A}\) is defined in Remark 4, satisfies 20 . The pairs \(\{q(z),I_p\}\) \(\break (z\in {\mathbb{C}}_+)\) such that \(q\) are \(p\times p\) holomorphic matrix functions and \[\begin{align} & \label{B343} q(z)^*q(z)\leq I_p, \end{align}\tag{23}\] are nonsingular, with property-\((-j)\) and all nonsingular pairs with property-\((-j)\) have the form \(\{q(z)a( z), a(z)\}\), where \(\det \big(a(z)\big)\not=0\). Since \(\break JK^*JKJ=-j\), the transformation \[\begin{align} & \label{B5} \begin{bmatrix}R(z) \\ Q(z)\end{bmatrix}=KJ\begin{bmatrix}q(z)a( z) \\ a( z)\end{bmatrix} \end{align}\tag{24}\] maps the set of nonsingular pairs with property-\((-j)\) onto the set of nonsingular pairs with property-\(J\).
Next, we compare linear fractional transformations 11 with linear fractional transformations from [7]: \[\begin{align} & \label{B4} f(\zeta)=\bigl( {{\mathcal{A}}}_{11}(\zeta)\widehat q(\zeta)+ {{\mathcal{A}}}_{12}(\zeta)\bigr) \bigl( {{\mathcal{A}}}_{21}(\zeta) \widehat q(\zeta)+{{\mathcal{A}}}_{22}(\zeta)\bigr)^{-1}, \end{align}\tag{25}\] where \({\mathcal{A}}_{ij}\) are \(p\times p\) blocks of \({\mathcal{A}}\) and \(\widehat q(\zeta)\) are contractive in \({\mathbb{D}}\). In view of 22 , 24 , the set \({\mathcal{N}}({\mathfrak A}(S))\) of matrix functions \(\varphi(z)\) given by 11 and the set \({\mathcal{N}}({\mathcal{A}})\) of matrix functions \(f(\zeta)\) given by 25 are connected by the relation \[\begin{align} & \label{B6} \widehat\varphi(\zeta)=\mathrm{i}f(\zeta) \quad {\mathrm{or}} \quad \varphi(z)=\mathrm{i}\breve f(z), \end{align}\tag{26}\] where \(\breve f\) is defined in ?? . It follows from the well-known Stieltjes-Perron formula (see, e.g., [15] and the bibliography there) that for the boundary values of \(\varphi\) and \(f\) considered in 26 we have \[\begin{align} & \label{B7} \Im (\varphi(t))=\Re\big(\breve f(t)\big)=\pi \mu^{\prime}(t) \quad (t\in {\mathbb{R}}), \end{align}\tag{27}\] where \(\mu\) is the nondecreasing matrix function in the Herglotz representation 13 of \(\varphi\).
Now, let us consider \(\chi(\zeta):=-{\mathcal{A}}_{22}(\zeta)^{-1}{\mathcal{A}}_{21}(\zeta)\). In view of 22 , we have \[\begin{align} & \label{B8} \chi(\zeta)=(\widehat{\mathfrak A}_{21}( \zeta)+\widehat{\mathfrak A}_{22}(\zeta))^{-1}(\widehat{\mathfrak A}_{21}( \zeta)-\widehat{\mathfrak A}_{22}( \zeta)), \end{align}\tag{28}\] where \(\widehat{\mathfrak A}_{ik}(\zeta)=\widehat{\mathfrak A}_{ik}(S,\zeta)\) are \(p\times p\) blocks of \(\widehat{\mathfrak A}(S,\zeta)\). Clearly, the requirement \[\begin{align} & \label{B1143} \|\chi(\zeta)\|<1 \quad {\mathrm{for}} \quad z\in {\mathbb{D}} \end{align}\tag{29}\] in [7] is equivalent to the condition \(\chi(\zeta)\chi(\zeta)^*<I_p\). Therefore, using 28 , after easy transformations we derive that 29 is equivalent to \[\begin{align} & \label{B9} \widehat{\mathfrak A}_{21}( \zeta)\widehat{\mathfrak A}_{22}(\zeta)^*+\widehat{\mathfrak A}_{22}( \zeta)\widehat{\mathfrak A}_{21}(\zeta)^*>0. \end{align}\tag{30}\] Under conditions 5 , the inequality 30 is immediate from 8 . Thus, the requirement 29 in [7] is fulfilled.
By virtue of 20 and 29 , the conditions of [7] are satisfied. Hence, the conditions of [7] are fulfilled for the case where the function \[\begin{align} & \label{B14} {\mathcal{A}}_{22}(\zeta)=\frac{1}{\sqrt{2}}\big(\widehat{\mathfrak A}_{21}( \zeta)+\widehat{\mathfrak A}_{22}(\zeta)\big) \end{align}\tag{31}\] is an outer matrix function. Moreover, taking into account 22 , in that case we have \[\begin{align} \nonumber \Delta(\zeta): &=\big({\mathcal{A}}_{22}(\zeta){\mathcal{A}}_{22}(\zeta)^*-{\mathcal{A}}_{21}(\zeta){\mathcal{A}}_{21}(\zeta)^*\big)^{-1} \\ & \label{B15} =\big(\widehat{\mathfrak A}_{21}( \zeta)\widehat{\mathfrak A}_{22}(\zeta)^*+\widehat{\mathfrak A}_{22}( \zeta)\widehat{\mathfrak A}_{21}(\zeta)^*\big)^{-1}, \end{align}\tag{32}\] where \(\Delta\) is an important notion from [7]. It follows from 8 , 15 and 32 that \[\begin{align} & \label{B1543} \Delta(\zeta)=\widehat\rho\big(\zeta, 1/ \, \overline{\zeta}\big)^{-1} \quad \big(\widehat\rho\big(\zeta, 1/ \, \overline{\zeta}\big)>0\big). \end{align}\tag{33}\]
The requirement above that \({\mathcal{A}}_{22}(\zeta)\) is an outer matrix function may be rewritten in the form \[\begin{align} & \label{B16} {\mathcal{A}}_{22}(\zeta), \, {\mathcal{A}}_{22}(\zeta)^{-1} \in D^{(p\times p)}, \end{align}\tag{34}\] where \(D^{(p\times \ell)}\) is introduced in Remark 5. Using 31 , we rewrite \({\mathcal{A}}_{22}(\zeta)\) as \[\begin{align} & \label{B17} {\mathcal{A}}_{22}(\zeta)=\frac{1}{\sqrt{2}}\widehat{\mathfrak A}_{21}( \zeta)\big(I_p+\widehat{\mathfrak A}_{21}( \zeta)^{-1}\widehat{\mathfrak A}_{22}(\zeta)\big). \end{align}\tag{35}\] Note that 30 yields the invertibility of \(\widehat{\mathfrak A}_{21}( \zeta)\) and, moreover, we have \(\Re\big(\widehat{\mathfrak A}_{21}( \zeta)^{-1}\widehat{\mathfrak A}_{22}(\zeta)\big)>0\). Now, it is immediate that \[\begin{align} & \label{B18} \Re\big(I_p+\widehat{\mathfrak A}_{21}( \zeta)^{-1}\widehat{\mathfrak A}_{22}(\zeta)\big)>0, \quad \Re\Big(\big(I_p+\widehat{\mathfrak A}_{21}( \zeta)^{-1}\widehat{\mathfrak A}_{22}(\zeta)\big)^{-1}\Big)>0. \end{align}\tag{36}\] Thus, according to Smirnov’s theorem (see, e.g., [12]) the functions \[\begin{align} & \label{B1843} h^*\big(I_p+\widehat{\mathfrak A}_{21}( \zeta)^{-1}\widehat{\mathfrak A}_{22}(\zeta)\big)h, \quad h^*\big(I_p+\widehat{\mathfrak A}_{21}( \zeta)^{-1}\widehat{\mathfrak A}_{22}(\zeta)\big)^{-1}h \end{align}\tag{37}\] belong to the classes \(H^{\delta}\) for any \(h\in {\mathbb{C}}^p\) and \(0<\delta<1\). Therefore, we have \[\big(I_p+\widehat{\mathfrak A}_{21}( \zeta)^{-1}\widehat{\mathfrak A}_{22}(\zeta)\big), \, \big(I_p+\widehat{\mathfrak A}_{21}( \zeta)^{-1}\widehat{\mathfrak A}_{22}(\zeta)\big)^{-1} \in D^{p\times p},\] and so it suffices that \[\begin{align} & \label{B19} \widehat{\mathfrak A}_{21}(\zeta), \, \widehat{\mathfrak A}_{21}(\zeta)^{-1} \in D^{(p\times p)} \end{align}\tag{38}\] for 34 to hold.
Summing up the reformulated conditions of [7], we state the result.
****Theorem** 8**. Let an \(S\)-node \(\{A,S, \Pi\}\), such that the operators \(I-zA^*\) have bounded inverses for \(z\in {\mathbb{C}}_+\) \((\)excluding, possibly, some isolated points, which are the poles of the matrix function \({\mathfrak A}(S,z))\), be given and let relations 5 and 38 hold. Assume that \(\varphi(z)\in {\mathcal{N}}({\mathfrak A}(S))\) and the matrix function \(\mu(t)\) in Herglotz representation 13 of \(\varphi\) satisfies Szegő condition 18 .
Then, we have \[\begin{align} & \label{B1333} 2 \pi G_{\mu}(z)^* G_{\mu}(z)\leq \rho\big(z, \overline{z}\big)^{-1} \quad (z \in {\mathbb{C}}_+). \end{align}\qquad{(5)}\] The equality in ?? holds in some point \(z=\lambda\in {\mathbb{C}}_+\) if and only if our \(\varphi(z)\in {\mathcal{N}}({\mathfrak A}(S))\) is generated \((\)see 11 \()\) by the constant pair \(\{R,Q\}:\) \[\begin{align} & \label{B31} R(z)\equiv {\mathfrak A}_{22}(S,\lambda)^*, \quad Q(z)\equiv {\mathfrak A}_{21}(S,\lambda)^*. \end{align}\qquad{(6)}\]
Proof. First note that \(\widehat{\mathfrak A}(S,\zeta) KJ\) satisfies conditions of [7] (as shown above). According to ?? and 27 , we have \[\begin{align} & \label{B12} \pi \widehat G_{\mu}(\zeta)^*\widehat G_{\mu}(\zeta)=\Re\big( f(\zeta)\big) \quad {\mathrm{for}} \quad |\zeta|=1. \end{align}\tag{39}\] Comparing 39 with the factorisation at the beginning of paragraph 3 in [7], we see that \(\sqrt{\pi}\widehat G_{\mu}(\zeta)\) coincides with \(\varphi_f(\zeta)\) in the notations of [7]. Thus, the inequality in [7] takes the form \[\begin{align} & \label{B13} 2 \pi \widehat G_{\mu}(\zeta)^*\widehat G_{\mu}(\zeta)\leq \Delta(\zeta) \quad (\zeta\in {\mathbb{D}}), \end{align}\tag{40}\] where the equality is attained at some \(\zeta=\widetilde{\zeta}\) if and only if \(\widehat q(\zeta)\equiv \chi(\widetilde{\zeta})^*\) (for \(\widehat q\) in 25 ). (We note that [7] contains a misprint and the coefficient “2" from eq. ¿eq:B13? is missing there.) Recall that the meaning of the accent”widehat" in the paper is explained in Remark 4. Hence, taking into account 33 , we see that the inequality 40 is equivalent to ?? . The interrelations between \(\widehat q\) and pairs \(\{R,Q\}\) in 11 , which generate \(\varphi(z)\) corresponding \(f(\zeta)\) (as in 27 ), is given by 24 . Therefore, condition \(\widehat q(\zeta)\equiv \chi(\widetilde{\zeta})^*\), where \(\chi(\zeta)\) has the form 28 , transforms into ?? . ◻
Relation 38 on \(\widehat{\mathfrak A}_{21}(\zeta)\) is the most complicated to check among the requirements stated in Theorem 8. Here, we present some sufficient conditions for 38 to hold. It follows from 2 that \[\begin{align} & \label{C0} {\mathfrak A}(S,z)=I_{2p}-\mathrm{i}z\Pi^*(I-zA^*)^{-1}S^{-1}\Pi J. \end{align}\tag{41}\] In particular, \({\mathfrak A}_{21}(S,z)\) has the form \[\begin{align} & \label{C1} c(z):={\mathfrak A}_{21}(S,z)=-\mathrm{i}z\Phi_2^*(I-zA^*)^{-1}S^{-1}\Phi_2. \end{align}\tag{42}\] Proposition 9 on \(\widehat c(\zeta)=\widehat{\mathfrak A}_{21}(\zeta)\) is helpful in the applications of Theorem 8.
****Proposition** 9**. Let a triple \(\{A,S,\Pi\}\) form an \(S\)-node and let the operators \(I-zA\) have bounded inverses for \(z\) in the domains \(\{z: \, \Im(z)\leq 0\}\) and \(\{z: \, \Im(z)>0, \,\, |z| \geq r_0\}\) for some \(r_0>0\). Assume that relations 5 hold. Finally, let \[\begin{align} & \label{Z2} {\overline{\lim}}_{r\to \infty}\big(\ln({\mathcal{M}}(r))\big/r^{\varkappa}\big)<\infty \end{align}\qquad{(7)}\] for some \(0<\varkappa<1\) and \({\mathcal{M}}(r)\) given by \[\begin{align} & \label{Z3} {\mathcal{M}}(r)=\sup_{r_0<|z|<r} \|(I-z A)^{-1}\|. \end{align}\qquad{(8)}\]
Then, we have \(\widehat c(\zeta), \, \widehat c(\zeta)^{-1} \in D^{(p\times p)}\), that is, \(\widehat c(\zeta)\) is an outer matrix function.
The proof of Proposition 9 consists of the proofs of two lemmas below.
****Lemma** 10**. Let a triple \(\{A,S,\Pi\}\) form an \(S\)-node, let the operator \(S\) have a bounded inverse, and let the operators \(I-z A^*\) have bounded inverses for \(\Im(z)\geq 0\). Assume that \[\begin{align} & \label{C5} {\overline{\lim}}_{r\to \infty}\big(\ln(M(r))\big/r^{\varkappa}\big)<\infty \quad {\mathrm{for\,\, some}} \quad 0<\varkappa<1, \end{align}\qquad{(9)}\] where \[\begin{align} & \label{C6} M(r)=\sup_{|z|<r, \,\, \Im(z)\geq 0} \|(I-z A^*)^{-1}\|. \end{align}\qquad{(10)}\] Then, \(\widehat c(\zeta)\in D^{(p\times p)}\).
Proof. It follows from 42 that \[\begin{align} & \label{C7} \|c(z)\| \leq {\mathcal{C}}_1 \, |z| \, \|(I-zA^*)^{-1}\| \quad {\mathrm{for \,\, some}} \quad {\mathcal{C}}_1>0. \end{align}\tag{43}\] According to Remark 7, it suffices to prove our lemma for the case \(z_0=\mathrm{i}\) in 16 . Thus, we set \[\begin{align} & \label{C8} z=- \mathrm{i}(\zeta+1)(\zeta-1) \end{align}\tag{44}\] Clearly, \(\widehat c(\zeta)\) is analytic on the closed unit disk \(|\zeta| \leq1\) excluding, possibly, the point \(\zeta=1\). By virtue of ?? –43 , we have \[\begin{align} & \label{C9} |\widehat c_{ij}(\zeta)|\leq {\mathcal{C}}_2 |\zeta-1|^{-1}\mathrm{e}^{{\mathcal{C}}_3 |\zeta-1|^{-\varkappa}} \end{align}\tag{45}\] for some \({\mathcal{C}}_2,{\mathcal{C}}_3\) and the entries \(\widehat c_{ij}\) of \(\widehat c\). We also note that \[\begin{align} & \label{C943} 2\big|r\mathrm{e}^{\mathrm{i}\vartheta}-1\big|>\big|\mathrm{e}^{\mathrm{i}\vartheta}-1\big| \quad (0<|\vartheta|<\pi, \quad 0<r<1). \end{align}\tag{46}\] In order to prove that \(\widehat c(\zeta)\in D^{(p\times p)}\), one needs to show (in the disk \(|\zeta|\leq 1\)) the equalities \[\begin{align} & \label{C10} \lim_{r \to 1}\int_{-\pi}^{\pi}\ln^{+}\big|\widehat c_{ij}\big(r\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta=\int_{-\pi}^{\pi}\ln^{+}\big|\widehat c_{ij}\big(\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta \end{align}\tag{47}\] for \(1\leq i,j \leq p\), where \(\ln^{+}a\) equals \(\ln a\) for \(a \geq1\) and equals \(0\) for \(a<1\). Equalities 47 follow from 45 , 46 and the continuity of \(\widehat c(\zeta)\) (excluding, possibly, the point \(\zeta=1\)) if we split the domain \([-\pi,\pi]\) of integration in 47 into the domains \([-\pi,\delta]\cup [\delta, \pi]\) and \((-\delta, \delta)\) with \(\delta\) tending to zero. In view of one of the equivalent definitions of the class \(D\) (see [12]), equalities 47 imply \(\widehat c(\zeta)\in D^{(p\times p)}\). ◻
****Remark** 11**. Similar to 46 , one obtains a slightly more general relation, which we will need further in the text. That is, for \(\vartheta,\vartheta_k\in {\mathbb{R}}, \,\, |\vartheta-\vartheta_k|<\pi\) we have \[\begin{align} & \label{C933} 2\big|r\exp\{\mathrm{i}\vartheta\}-\exp\{\mathrm{i}\vartheta_k\}\big|>\big|\exp\{\mathrm{i}\vartheta\}-\exp\{\mathrm{i}\vartheta_k\}\big| . \end{align}\qquad{(11)}\]
****Remark** 12**. We note that the requirement of the invertibility of \((I-z A^*)\) for \(\Im(z)\geq 0\) does not hold for our standard triple \(\{A,S,\Pi\}\) corresponding to Toeplitz matrices \(S\) \((\)see [6]\()\). However, this triple is easily substituted by the triple \(\{\widetilde{A},S, \widetilde{\Pi}\}\), where \(\widetilde{A}=-A\) and \(\widetilde{\Pi}=\begin{bmatrix} -\Phi_2 & \Phi_1\end{bmatrix}\) and the corresponding asymptotic result is obtained in this way in [5].
Let the conditions of Lemma 10 and relations 5 be satisfied. Then, we easily see that \(c(z)\) is analytic in the domains \[\begin{align} & \label{Z1} \{z: \, -\widetilde{r} \leq \Re(z)\leq \widetilde{r}, \,\, \Im(z)\geq -\varepsilon(\widetilde{r})\} \end{align}\tag{48}\] for all the values \(\widetilde{r} >0\) and some values \(\varepsilon(\widetilde{r})>0\). Since \(S^{-1}>0\) and \({\rm Ker}\, \Phi_2=~0\), similar to 8 , we obtain \[\begin{align} & \label{I8-} {\mathfrak A}_{21}(z){\mathfrak A}_{22}(z)^*+{\mathfrak A}_{22}(z){\mathfrak A}_{21}(z)^*<0 \quad {\mathrm{for}}\quad \Im(z) <0. \end{align}\tag{49}\] Formulas 8 and 49 imply that \(c(z)={\mathfrak A}_{21}(S,z)\) is invertible in these domains excluding possibly some values \(z\in {\mathbb{R}}\). This yields that \(\det \big(c(z)\big)\) may have only finite number of zeros of finite order in each domain 48 and these zeros are placed on the interval \(-\widetilde{r}<z<\widetilde{r}\).
We again assume the correspondence 44 between \(z\) and \(\zeta\). Recall that \(\widehat c(\zeta)\in D^{(p\times p)}\) and so \(\det \widehat c(\zeta)\in D\) and the minors of \(\widehat c(\zeta)\) belong \(D\) as well. Since the functions from \(D\) may be represented as ratios of bounded functions analytic in the unit disk, the entries of \(\widehat c(\zeta)^{-1}\) may be represented as such ratios as well. Moreover \(\widehat c(z)^{-1}\) is analytic in the unit disk because \(c(z)\) is invertible in \({\mathbb{C}}_+\). Thus, the entries of \(\widehat c(\zeta)^{-1}\) have characteristic properties of the functions from class \({\boldsymbol{A}}\) (see [12]), that is, \(\widehat c(\zeta)^{-1}\in {\boldsymbol{A}}^{(p\times p)}\). For the entries \(\big(\widehat c(\zeta)^{-1}\big)_{ij}\) of \(\widehat c(\zeta)^{-1}\), it follows (see [12]) that \[\begin{align} & \label{C3} \lim_{r\to 1}\int_{-\pi}^{\pi}\ln^{+}\big|\big(\widehat c(r\mathrm{e}^{\mathrm{i}\vartheta})^{-1}\big)_{ij}\big|d\vartheta<\infty , \end{align}\tag{50}\] and (see [12]) that \[\begin{align} & \label{C343} \int_{-\pi}^{\pi}\ln^{+}\big|\big(\widehat c(\mathrm{e}^{\mathrm{i}\vartheta})^{-1}\big)_{ij}\big|d\vartheta<\infty . \end{align}\tag{51}\] However, relations 50 and 51 are insufficient for our purposes. In order to show that \[\begin{align} & \label{C4} \lim_{r\to 1}\int_{-\pi}^{\pi}\ln^{+}\big|\big(\widehat c(r\mathrm{e}^{\mathrm{i}\vartheta})^{-1}\big)_{ij}\big|d\vartheta=\int_{-\pi}^{\pi}\ln^{+}\big|\big(\widehat c(\mathrm{e}^{\mathrm{i}\vartheta})^{-1}\big)_{ij}\big|d\vartheta, \end{align}\tag{52}\] that is, \(\widehat c(\zeta)^{-1} \in D^{(p\times p)}\), we need stronger conditions.
****Lemma** 13**. Let a triple \(\{A,S,\Pi\}\) form an \(S\)-node and let the operators \(I-zA^*\) have bounded inverses for \(z\) in the domain \(\Im(z)\geq 0\) as well as for \(z\) in the domain \(\{z: \, \Im(z)<0, \,\, |z| \geq r_0\}\) for some \(r_0>0\). Assume that relations 5 and ?? hold. Finally, let \[\begin{align} & \label{C239} {\overline{\lim}}_{r\to \infty}\big(\ln(\widetilde{M}(r))\big/r^{\varkappa}\big)<\infty \quad {\mathrm{for\,\, some}} \quad 0<\varkappa<1, \end{align}\qquad{(12)}\] where \[\begin{align} & \label{C339} \widetilde{M}(r)=\sup_{\Im(z)\leq 0, \, r_0<|z|<r} \|(I-z A^*)^{-1}\|. \end{align}\qquad{(13)}\]
Then, we have \(\widehat c(\zeta), \, \widehat c(\zeta)^{-1} \in D^{(p\times p)}\).
Proof. Since the conditions of Lemma 10 are satisfied, we see that \(\break \widehat c(\zeta) \in D^{(p\times p)}\). In order to show that \(\widehat c(\zeta)^{-1} \in D^{(p\times p)}\), we need some preparations.
Inequality 7 yields \[\begin{align} & \label{C11} \begin{bmatrix} a(z)^* & c(z)^*\end{bmatrix}J \begin{bmatrix} a(z) \\ c(z)\end{bmatrix}\geq 0 \quad \big(a(z):={\mathfrak A}_{11}(S,z)\big). \end{align}\tag{53}\] Formula 53 implies that \(a(z)c(z)^{-1}+ \big(c(z)^{-1}\big)^*a(z)^*\geq 0\). Now, using Smirnov’s theorem similar to the considerations for the functions 37 , we obtain \[\begin{align} & \label{C12} \Upsilon(\zeta)=\{\Upsilon_{i,j}\}_{i,j=1}^p(\zeta):=\widehat a(\zeta) \widehat c(\zeta)^{-1}\in D^{(p \times p)}. \end{align}\tag{54}\]
By virtue of 6 , we have \[\begin{align} & \label{C13} {\mathfrak A}(S,z)^{-1}=J{\mathfrak A}(S,\overline{z})^*J. \end{align}\tag{55}\] Recall that \(a={\mathfrak A}_{11}\) and \(c={\mathfrak A}_{21}\). We will use the following representation: \[\label{C14} c(z)^{-1}=\begin{bmatrix} I_p & 0\end{bmatrix}{\mathfrak A}(S,z)^{-1}\begin{bmatrix} a(z) \\ c(z)\end{bmatrix}c(z)^{-1}={\mathcal{V}}(z) \begin{bmatrix} a(z)c(z)^{-1} \\ I_p\end{bmatrix},\tag{56}\] where \[\begin{align} & \label{C15} {\mathcal{V}}(z):=\begin{bmatrix} I_p & 0\end{bmatrix}{\mathfrak A}(S,z)^{-1}=\begin{bmatrix} I_p & 0\end{bmatrix}J{\mathfrak A}(S,\overline{z})^*J. \end{align}\tag{57}\] In view of 54 and 56 , it suffices to show that \(\widehat{\mathcal{V}}(\zeta)\) has a good behaviour near the unit circle in order to prove this proposition.
Indeed, it follows from 2 and 57 that \[\begin{align} & \label{C16} {\mathcal{V}}(z)=\begin{bmatrix} I_p & 0\end{bmatrix}+ \mathrm{i}z \Phi_1^*S^{-1}(I-zA)^{-1}\Pi J. \end{align}\tag{58}\] Similar to 45 , one can show that \[\begin{align} & \label{C17} |\widehat{\mathcal{V}}_{ij}(\zeta)|\leq {\mathcal{C}}_4 |\zeta-1|^{-1}\mathrm{e}^{{\mathcal{C}}_5 |\zeta-1|^{-\varkappa}} \quad (1\leq i\leq p, \,\, 1\leq j \leq 2p) \end{align}\tag{59}\] in some open neighbourhood of \(\zeta=1\) in the disk \(|\zeta| \leq 1\). Let us fix a set \[\Omega=\Omega(\widetilde{r}, \widetilde{\delta})=\{(\delta,r): \, 0<\delta\leq \widetilde{\delta}, \,\, 0<\widetilde{r}\leq r\leq1\},\] where 59 holds for all \(\zeta= r\mathrm{e}^{\pm \mathrm{i}\delta}\) such that \((\delta,r)\in \Omega\). From 46 , 56 and 59 we derive (for \((\delta,r)\in \Omega\)) that \[\begin{align} \nonumber \int_{-\delta}^{\delta}\ln^{+}\big|\big(\widehat c(r\mathrm{e}^{\mathrm{i}\vartheta})^{-1}\big)_{i,j}\big|d\vartheta\leq & \int_{-\delta}^{\delta}\ln^{+}\big( {\mathcal{C}}_6 \big|\mathrm{e}^{\mathrm{i}\vartheta}-1\big|^{-1}\exp\{{\mathcal{C}}_7 \big|\mathrm{e}^{\mathrm{i}\vartheta} -1|^{-\varkappa}\}\big)d\vartheta \\ & \label{C18} +\sum_{k=1}^p\int_{-\delta}^{\delta}\ln^{+}\big|\Upsilon_{kj}\big(r\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta+ 2\delta\ln(p+1). \end{align}\tag{60}\]
Let us consider \(\Upsilon(\zeta)\) given in 54 . It follows from 41 that \(\widehat a(\zeta)\) and \(\widehat c(\zeta)\) are holomorphic on the disk \(|\zeta| \leq 1\) excluding, possibly, the point \(\zeta=1\). Using our considerations on \(\det\big(c(z)\big)\) before this lemma, we see that \(\widehat c(\zeta)^{-1}\) is holomorphic on the disk \(|\zeta| \leq 1\) excluding, possibly, the point \(\zeta=1\) and poles on the unit circle. Therefore, \(\Upsilon(\zeta)\) is also holomorphic on the disk \(|\zeta| \leq 1\) excluding, possibly, the point \(\zeta=1\) and poles on the unit circle. Moreover, there is a finite number of these poles \(\zeta=\exp\{\mathrm{i}\vartheta_{\ell}\}\) on the arc \(\zeta=\mathrm{e}^{\mathrm{i}\vartheta}\), where \(|\vartheta| \geq \delta\). Hence, using ?? , we see that \[\begin{align} & \tag{61} \int_{-\pi}^{\delta}\ln^{+}\big|\Upsilon_{kj}\big(r\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta\to \int_{-\pi}^{\delta}\ln^{+}\big|\Upsilon_{kj}\big(\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta, \\ & \tag{62} \int_{\delta}^{\pi}\ln^{+}\big|\Upsilon_{kj}\big(r\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta\to \int_{\delta}^{\pi}\ln^{+}\big|\Upsilon_{kj}\big(\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta, \end{align}\] for \(1\leq k \leq p, \,\, r\to 1\). We note that 61 and 62 are derived by considering separately the intervals of integration \((\vartheta_{\ell}-\delta_{\ell},\vartheta_{\ell}+\delta_{\ell})\) (\(\delta_{\ell}\to 0\)) in order to exclude the poles. Since \(\Upsilon_{kj}(\zeta)\in D\), formulas 61 and 62 imply a similar relation on the arc \(|\vartheta|\leq \delta\): \[\begin{align} & \label{C21} \int_{-\delta}^{\delta}\ln^{+}\big|\Upsilon_{kj}\big(r\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta\to \int_{-\delta}^{\delta}\ln^{+}\big|\Upsilon_{kj}\big(\mathrm{e}^{\mathrm{i}\vartheta}\big)\big|d\vartheta. \end{align}\tag{63}\] It follows from 60 and 63 that for any \(\varepsilon>0\) there are a sufficiently small \(\delta\) (\(\delta\leq \widetilde{\delta}\)) and a value \(r_{\varepsilon}\) \((\widetilde{r} \leq r_{\varepsilon}< 1)\) such that for \(r\) satisfying \(r_{\varepsilon}\leq r\leq 1\) we have \[\begin{align} & \label{C22} \int_{-\delta}^{\delta}\ln^{+}\big|\big(\widehat c(r\mathrm{e}^{\mathrm{i}\vartheta})^{-1}\big)_{i,j}\big|d\vartheta< \varepsilon/4 . \end{align}\tag{64}\] Similar to the proof of 61 and 62 , we obtain that (for the fixed \(\delta\) discussed above, some \(\widetilde{r}_{\varepsilon}\) \((\widetilde{r} \leq \widetilde{r}_{\varepsilon}<1)\), and all \(r\) such that \(\widetilde{r}_{\varepsilon} \leq r <1\)) the inequality \[\label{C23} \Big| \int_{\delta\leq |\vartheta|\leq \pi}\ln^{+}\big|\big(\widehat c(\mathrm{e}^{\mathrm{i}\vartheta})^{-1}\big)_{i,j}\big|d\vartheta-\int_{\delta\leq |\vartheta|\leq \pi}\ln^{+}\big|\big(\widehat c(r\mathrm{e}^{\mathrm{i}\vartheta})^{-1}\big)_{i,j}\big|d\vartheta\Big|<\varepsilon/2\tag{65}\] holds. Taking into account that some \(\delta\), \(r_{\varepsilon}\) and \(\widetilde{r}_{\varepsilon}\) exist for each \(\varepsilon\), we see that the inequalities 64 and 65 imply 52 , that is, \(\big(\widehat c(\zeta)^{-1}\big)_{i,j}\in D\). Thus, we have \(\widehat c(\zeta)^{-1}\in D^{(p\times p)}\). ◻
Proposition 9 follows from Lemma 13 after minor changes including formulation of the conditions in terms of \(I-zA\) instead of \(I-zA^*\).
This research was supported by the Austrian Science Fund (FWF) grant, DOI: 10.55776/Y963.↩︎