Preamble

Functional Analysis and Its Applications, Vol. 2, No. 1, 1968, 1–19.

On Infinite Hankel Matrices and Generalized Carathéodory–Fejér and F. Riesz Problems

V. M. Adamyan, D. Z. Arov, and M. G. Krein

Introduction

In this paper, we investigate the properties of infinite Hankel matrices and their relationship to classical interpolation problems in the theory of functions. Specifically, we consider the generalized Carathéodory–Fejér and F. Riesz problems, which involve finding bounded analytic functions in the unit disk that satisfy certain interpolation conditions or approximate given power series coefficients.

The study of Hankel matrices has long been connected to the theory of moments and the approximation of functions. A central role in our analysis is played by the singular values and vectors of these matrices. We aim to provide a unified framework for understanding how the spectral properties of a Hankel operator relate to the existence and uniqueness of solutions to these extremal problems.

1. Basic Definitions and Notation

Let $\mathcal{H}$ denote the Hilbert space of sequences $\xi = \{\xi_k\}_{0}^{\infty}$ such that $\sum |\xi_k|^2 < \infty$. An infinite Hankel matrix is defined by a sequence of complex numbers $\{c_k\}_{1}^{\infty}$ such that the entry in the $i$-th row and $j$-th column is $c_{i+j-1}$. We denote such a matrix by $\Gamma = (c_{i+j-1})_{i,j=1}^{\infty}$.

We are primarily interested in bounded Hankel operators acting on $\mathcal{H}$. According to Nehari's theorem, a Hankel matrix $\Gamma$ defines a bounded operator if and only if there exists a bounded measurable function $f(\zeta)$ on the unit circle $| \zeta | = 1$ such that its Fourier coefficients with negative indices are given by:
$$c_k = \frac{1}{2\pi} \int_{0}^{2\pi} f(e^{i\theta}) e^{ik\theta} d\theta, \quad k=1, 2, \dots$$
In this case, the norm of the operator $\|\Gamma\|$ is equal to the distance from $f$ to the space $H^\infty$ in the $L

Introduction

For a summable function $f(\zeta)$ defined on the unit circle $\mathbb{T}$ ($\zeta = e^{i\theta}$, $0 \leq \theta < 2\pi$), let $c_k(f) = \frac{1}{2\pi i} \int_{\mathbb{T}} f(\zeta) \zeta^{-k-1} d\zeta$ ($k = 0, \pm 1, \pm 2, \dots$) denote its Fourier coefficients, where the integral over the circle is taken in the positive direction and $d\zeta = i\zeta d\theta$. We are primarily interested in the following problem:

Problem A. Given a non-zero sequence of complex numbers $\{\gamma_k\}_{1}^{\infty}$, find, among the set $\mathfrak{M}(\Gamma)$ of all bounded functions $f \in L_\infty$ for which $c_k(f) = \gamma_k$ ($k = 1, 2, \dots$), a function that deviates least from zero in the $L_\infty$ metric.

It is relatively simple to show that the set $\mathfrak{M}(\Gamma)$ is non-empty if and only if the Hankel matrix $\Gamma = (\gamma_{j+k-1})_{j,k=1}^{\infty}$ is bounded (i.e., the operator defined by the matrix $\Gamma$ in the Hilbert space $l_2 = \{ \xi = \{\xi_j\}_1^\infty : \sum |\xi_j|^2 < \infty \}$ is bounded; this operator will be denoted by the same letter $\Gamma$). When this condition is satisfied, Problem A always has a solution, and the norm of the required function is equal to the norm of the operator $\Gamma$ [**]. It is easy to see that Problem A is a generalization of the classical Carathéodory–Fejér problem \cite{2}.

[] $L_p$ ($1 \leq p \leq \infty$) denotes the Banach space of measurable functions $f(\zeta)$ ($\zeta \in \mathbb{T}$) with the norm defined for $p < \infty$ by $\|f\|_p = (\frac{1}{2\pi} \int_0^{2\pi} |f(e^{i\theta})|^p d\theta)^{1/p}$, and for $p = \infty$ by $\|f\|_\infty = \text{ess sup} |f(\zeta)|$. To obtain these results, one only needs to add a little to the arguments found, for example, in \cite{1} or \cite{17}. Note added in proof:* These results also appear in \cite{18}.

Problem $A_n$. Given complex numbers $a_1, a_2, \dots, a_n$, find among the functions $f(z) = \sum_{k=n}^{\infty} \gamma_k z^{-k} + \dots$ [sic] regular in the disk $|z| < 1$, a function that deviates least from zero in the $H_\infty$ metric [*]. In a certain sense, the dual to Problem $A_n$ is the problem of F. Riesz \cite{4}:

Problem $B_n$. Given complex numbers $a_1, a_2, \dots, a_n$, find among the functions $g(z) = b_0 + b_1 z + \dots$ regular in the disk $|z| < 1$ satisfying the condition

$$a_1 b_0 + a_2 b_1 + \dots + a_n b_{n-1} = 1,$$

a function that deviates least from zero in the $H_1$ metric. We omit the exact formulation of Problem B, which is a generalization of Problem $B_n$ to the case where an infinite sequence of numbers $a_1, a_2, \dots$ is given. This is contained in Theorems 1.2, 2.1, and 2.2, which simultaneously provide a complete solution to Problem B.

Regarding Problem A, this article mainly studies the case where the Hankel matrix $\Gamma$ has an attainable norm []. Incidentally, when this condition is met, Problem A has a unique solution $f$ and $|f(\zeta)| = \text{const}$ ($|\zeta| = 1$). This case occurs, for example, whenever the matrix $\Gamma$ is completely continuous (i.e., the corresponding operator in $l_2$ is compact), particularly when the sequence $\{\gamma_j\}_1^\infty$ is finite or $\gamma_j \to 0$ sufficiently rapidly. It is easy to see that Problem $A_n$ is equivalent to Problem A with a finite sequence $\{\gamma_j\}_1^n$. It turns out (Theorem 3.2) that if the matrix is completely continuous, then and only then is there a continuous function for which $\gamma_j$ ($j=1, 2, \dots$) serve as Fourier coefficients. Nevertheless, the solution to Problem A in this case is not necessarily a continuous function. If, however, the series $\sum |\gamma_j|$ converges, then the series composed of all Fourier coefficients of the solution $f$ to Problem A also converges absolutely (Theorem 4.1), and consequently, $f \in C$ [*]. This result generalizes to the case where the desired solution must belong to a particular Banach algebra or class of analytic functions on the unit circle (§ 4).

Problem A for the case where the matrix $\Gamma$ has an unattainable norm will be considered elsewhere. In that case, Problem A may be determinate (the solution is unique) or indeterminate (the solution is not unique). A criterion for determinacy and a description of all solutions will be provided.

[*] $H_p$ ($1 \leq p \leq \infty$) denotes the subspace of functions in $L_p$ for which $c_k(f) = 0$ for $k < 0$. The subspace $H_p$ will be identified with the Hardy space of functions $f(z)$ regular in the disk $|z| < 1$, for which the norm is defined for $p < \infty$ by $\|f\|_p = \sup_{r < 1} (\frac{1}{2\pi} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta)^{1/p}$, and for $p = \infty$ by

$$\|f\|_\infty = \sup_{|z| < 1} |f(z)|.$$

This identification is achieved by the correspondence between regular functions of the Hardy class and their boundary values \cite{3}.
[] That is, $\|\Gamma\|^2$ is an eigenvalue of the matrix $\Gamma^* \Gamma$.
[*] $C$ denotes the subspace of all continuous functions in $L_\infty$.

...in the indefinite case. Furthermore, for any $\rho > \|\Gamma\|$, we obtain a description of all functions $f$ from $M(\Gamma)$ for which $\|f\|_{\infty} < \rho$. This result can be viewed as a development of the well-known studies by I. Schur \cite{5} in a new direction. The authors also succeeded in generalizing the result of T. Takagi \cite{6} (see also \cite{7}), which allowed, in particular, for a complete approximation characterization of all $s$-numbers of a completely continuous matrix $\Gamma$.

It should be noted that the method we employ to obtain the aforementioned results allows for their generalization to matrix-valued functions and, to a significant extent, to functions whose values are operators in a Hilbert space. All these results have continuous analogs in the theory of Fourier integrals. In the present article, we are unable to dwell on the connections between the issues under consideration and various problems in scattering theory (and perturbation theory of operators in general), as well as problems involving the factorization of scalar and matrix functions.

During the course of this work, we received a preprint of an article by H. Helson and D. Sarason \cite{8}. This preprint proved to be extremely useful, and we express our gratitude to its authors.

§ 1. Starting Theorem

Let $\{\gamma_j\}_1^\infty$ be a non-zero sequence of complex numbers such that $\lim_{j \to \infty} \gamma_j = 0$. With this sequence, we associate the following objects:
1) the Hankel matrix $\Gamma = (\gamma_{j+k-1})_{j,k=1}^\infty$;
2) the set $M(\Gamma)$ of all $f \in L_\infty$ for which $c_{-j}(f) = \gamma_j$ ($j = 1, 2, \dots$); (1.1)
3) the set $H_\Gamma$ of all $g \in H_1^0$ for which the series $\sum_{j=1}^\infty \gamma_j c_{-j}(g)$ is $1$-summable to a finite sum $\Phi_\Gamma(g)$, where
$$\Phi_\Gamma(g) = \lim_{r \to 1^-} \sum_{j=1}^\infty \gamma_j c_{-j}(g) r^j;$$ (1.2)
4) the functional $\Phi_\Gamma$ defined on $H_\Gamma$ by the expression (1.2).
To each vector $\xi = \{\xi_j\}_1^\infty \in l_2$, we associate functions from $H_2$ and $H_2^\perp = L_2 \ominus H_2$:
$\xi_+(\zeta) = \sum_{j=1}^\infty \xi_j \zeta^{j-1}$ and $\xi_-(\zeta) = \sum_{j=1}^\infty \xi_j \zeta^{-j}$ ($|\zeta|=1$).
The mapping $\xi \to \xi_+$ ($\xi \to \xi_-$) is an isometric mapping of the entire space $l_2$ onto $H_2$ ($H_2^\perp$). Obviously, $\|\xi\|_2 = \|\xi_+\|_2 = \|\xi_-\|_2$.
*) A number $s_j > 0$ is called an $s$-number of a completely continuous operator $\Gamma$ if $s_j^2$ is an eigenvalue of the operator $\Gamma^* \Gamma$.

Theorem 1.1. For a non-zero sequence $\{\gamma_j\}_1^\infty$, the following conditions are equivalent:
I) the set $M(\Gamma)$ is non-empty;
II) the Hankel matrix $\Gamma = (\gamma_{j+k-1})_{j,k=1}^\infty$ is bounded;
III) the functional $\Phi_\Gamma$ is continuous on $H_1^0 \cap \mathcal{P}$ (where $\mathcal{P}$ is the set of polynomials);
IV) $H_\Gamma = H_1^0$.
If any of these conditions are satisfied, then
$$\sup \{ |\Phi_\Gamma(g)| / \|g\|_1 : g \in H_1^0 \} = \|\Gamma\| = \min \{ \|f\|_\infty : f \in M(\Gamma) \}.$$ (1.3)

Proof. Let $M(\Gamma)$ be non-empty; i.e., (1.1) holds for some $f \in L_\infty$. It is elementary to verify that for such an $f$ and any finite vectors $\xi, \eta \in l_2$, the following relations hold:
$$(\Gamma \xi, \eta) = \frac{1}{2\pi} \int_0^{2\pi} f(e^{it}) \xi_+(e^{it}) \eta_-(e^{it}) dt,$$ (1.4)
$$|(\Gamma \xi, \eta)| \le \frac{1}{2\pi} \int_0^{2\pi} |f(e^{it})| |\xi_+(e^{it})| |\eta_-(e^{it})| dt \le \|f\|_\infty \|\xi_+\|_2 \|\eta_-\|_2 = \|f\|_\infty \|\xi\|_2 \|\eta\|_2.$$
Consequently, $\|\Gamma\| \le \|f\|_\infty$ for each $f \in M(\Gamma)$. Thus, I) implies II).
We now show that II) implies all other statements, including the equalities in (1.3). Since $\|\Gamma\| < \infty$, the series $\sum \gamma_{j+k-1} \xi_j \eta_k$ converges, and therefore for any $g \in H_1^0 \cap \mathcal{P}$, by Parseval's identity, we have $\sum \gamma_j c_{-j}(g) = \Phi_\Gamma(g)$. Let $g = b \phi^2$ be the canonical representation of the function $g \in H_1^0$.*) Then, setting $\xi_+ = b \phi$ and $\eta_- = \bar{\phi}$, we obtain

$$\Phi_\Gamma(g) = \frac{1}{2\pi} \int_0^{2\pi} f(e^{it}) \xi_+(e^{it}) \eta_-(e^{it}) dt = (\Gamma \xi, \eta).$$

Since $\|\xi_+\|_2^2 = \|\eta_-\|_2^2 = \|\phi\|_2^2 = \|g\|_1$, it follows that
$$|\Phi_\Gamma(g)| \le \|\Gamma\| \|\xi\|_2 \|\eta\|_2 = \|\Gamma\| \|g\|_1.$$ (1.5)
The estimate (1.5) shows that the functional $\Phi_\Gamma$ is defined and continuous on the subspace $H_1^0 \cap \mathcal{P}$, which is dense in $H_1^0$.
*) We call the representation $g = b \phi^2$ for a function $g \in H_1^0$ canonical, where $b$ is an inner function ($b \in H_\infty$ and $|b(\zeta)| = 1$ almost everywhere for $|\zeta| = 1$) and $\phi \in H_2$ is an outer function. If $\phi$ is normalized by the condition $\phi(0) > 0$, the representation is unique (see, for example, \cite{3}). For a function $h \in H_1$, the canonical representation is of the form $b \phi^2$, where $b$ and $\phi$ have the same meaning as before.

Based on the Hahn-Banach theorem, we extend the restriction of the linear functional $\Phi_\Gamma$ from $H_1^0 \cap \mathcal{P}$ to the entire space $L_1$ while preserving the norm. Let this extended functional be $\Phi$, such that
$$\sup \{ |\Phi_\Gamma(g)| / \|g\|_1 : g \in H_1^0 \cap \mathcal{P} \} = \|\Phi\| = \sup \{ |\Phi(g)| / \|g\|_1 : g \in L_1 \}.$$ (1.6)
According to the well-known Riesz representation theorem, the general form of a continuous linear functional $\Phi$ on $L_1$ is given by the formula
$$\Phi(g) = \Phi_f(g) = \frac{1}{2\pi} \int_0^{2\pi} f(e^{it}) g(e^{it}) dt,$$ (1.7)
where $f$ is some function in $L_\infty$ and $\|\Phi\| = \|f\|_\infty$. Applying this to $\Phi = \Phi_\Gamma$, the last equality together with (1.5) and (1.6) yields $\|f\|_\infty \le \|\Gamma\|$. On the other hand, since $\Phi(g) = \Phi_\Gamma(g)$ for $g \in H_1^0 \cap \mathcal{P}$, we have in particular
$$c_{-j}(f) = \Phi_\Gamma(\zeta^{-j}) = \gamma_j \quad (j = 1, 2, \dots).$$ (1.8)
Thus, $f \in M(\Gamma)$, and by the previously proven implication (I $\implies$ II), we have $\|f\|_\infty \ge \|\Gamma\|$, which implies $\|f\|_\infty = \|\Gamma\|$. This proves that II) implies I) and the right-hand equality in (1.3).
Next, we show that $H_\Gamma = H_1^0$. As is well known (see \cite{3}, p. 42), for every $g \in H_1^0$, the Fejér sums $\sigma_n(\zeta; g) = \sum_{j=1}^n (1 - \frac{j}{n+1}) c_{-j}(g) \zeta^{-j}$ ($n = 1, 2, \dots$) converge to $g$ in the $H_1$ metric. It then follows that
$$\Phi_\Gamma(g) = \lim_{n \to \infty} \Phi_\Gamma(\sigma_n).$$
According to (1.2), the existence of this limit means that $g \in H_\Gamma$. Thus, indeed $H_\Gamma = H_1^0$. Simultaneously, we have shown that $\Phi_\Gamma(g) = \Phi_f(g)$ for $g \in H_1^0$. This also proves III).
If III) and IV) hold, then in (1.6) one can replace $H_1^0 \cap \mathcal{P}$ with $H_1^0$, which yields the left-hand equality in (1.3), since we have already shown $\|\Phi\| = \|\Phi_\Gamma\| = \|\Gamma\|$.
Thus, it has been shown that II) implies all statements of the theorem. Analyzing the preceding arguments, we note that statement I) follows solely from the continuity of the restriction of the functional $\Phi_\Gamma$ to the set of all polynomials $g(\zeta) = \sum c_{-k}(g) \zeta^{-k}$. Thus, III) implies I), and consequently all other statements. It remains to show that IV) implies III). Statement IV) implies that the functionals $\Phi_N$ ($N = 1, 2, \dots$):

<\g)=^ (i-A^r.+ic_,(g^)

converge on the entire space $H_i$. Since $H_i$ is complete and $\Phi_i^{(N)}$ ($N = 1, 2, \dots$) are continuous functionals on $H_i$, their weak limit $\Phi_i$ is also continuous.

Remark

1.1. We can now expand the content of the used machine learning and deep learning methods to improve the accuracy of the proposed model.

derived formulas and state the following proposition.

Let $\| \Gamma \| < \infty$ and $f \in L_{\infty}(\Gamma)$. Then

$$\Phi_f(g) = \Phi_f(g(e^{it})) = \int_{\Gamma} g(e^{it}) f(e^{it}) dt \quad (g \in H_1)$$ (1.9)
and for any $\eta \in L_2(\Gamma)$,
$$\|\Gamma \eta\|_2 = \sup_{g \in H_2, \|g\|_2=1} |\Phi_f(\eta g)| = \|\Phi_f(\eta \cdot)\|.$$ (1.10)
Both equalities are obtained by a simple application of Parseval's formula: in the second case to the functions $f \eta$ and $\bar{g}$, and in the first case to the functions $f$ and $g$. In the latter case, it is known \cite{9} that the series composed of the products of the corresponding Fourier coefficients of the functions is summable by the $(C, 1)$ method. If $f \in L_{\infty}$, we shall denote the Hankel matrix $(c_{j+k-1}(f))_{j,k=1}^{\infty}$ by $\Gamma(f)$. The conditions (1.1) will now be written briefly as $\Gamma(f) = \Gamma$. According to the right-hand equality in (1.3), for $f \in L_{\infty}$ we have $\|\Gamma(f)\| \leq \|f\|_{\infty}$.

Definition. A function $f \in L_{\infty}$ is called a minifunction if $\|f\|_{\infty} = \|\Gamma(f)\|$. By Nehari's theorem, every bounded Hankel matrix corresponds to at least one minifunction such that $\Gamma(f) = \Gamma$.

Corollary 1. Let $\Gamma_n = \Gamma(z^{-n}f)$ ($n = 1, 2, \dots$). For $f \in L_{\infty}$, it is necessary and sufficient that the matrices $\Gamma_n$ have norms bounded by the same constant. When this condition is met, $\|\Gamma_n\| \leq \|\Gamma_{n+1}\| \leq \|f\|_{\infty} = \lim_{n \to \infty} \|\Gamma_n\|$.

Proof. Indeed, if $f \in L_{\infty}$, then according to (1.3), $\|\Gamma_n\| \leq \|z^{-n}f\|_{\infty} = \|f\|_{\infty}$. If $\Gamma_n$ is a bounded matrix, then $\Gamma_{n+1}$ will also be bounded, since $\Gamma_n$ is obtained from $\Gamma_{n+1}$ by adding one row; for this reason, $\|\Gamma_n\| \leq \|\Gamma_{n+1}\|$ ($n = 1, 2, \dots$). Let $f_n$ be a minifunction such that $\Gamma(z^{-n}f) = \Gamma(f_n)$ and $\|f_n\|_{\infty} = \|\Gamma_n\|$ ($n = 1, 2, \dots$). Let $f^{(n)} = z^n f_n$; then $\|f^{(n)}\|_{\infty} = \|\Gamma_n\|$ and $\Gamma_n = \Gamma(z^{-n}f^{(n)})$. This last equality implies that $\Gamma(z^{-n}f^{(n)}) = \Gamma(z^{-n}f)$, i.e.,

$$c_k(f^{(n)}) = c_k(f) \quad (k = -n+1, -n+2, \dots).$$ (1.11)
To each $f_n$, according to formula (1.7), there corresponds a linear continuous functional $\Phi_n$ ($n = 1, 2, \dots$) such that $\|\Phi_n\| = \|f_n\|_{\infty} = \|\Gamma_n\| < \infty$. According to (1.11), for any $g$ in the linear span of all powers $z^k$ ($k = 0, \pm 1, \pm 2, \dots$), there exists a limit $\lim_{n \to \infty} \Phi_n(g) = \Phi(g)$. Since $\|\Phi_n\| \leq A$, the limiting linear functional has norm $\|\Phi\| \leq A$. By continuity, it is defined on the closure $L_1$ while preserving the norm. Let $f$ be the corresponding function from $L_{\infty}$ such that $\Phi = \Phi_f$, so that $\|f\|_{\infty} \leq A$. According to (1.11), we have $c_k(f) = \lim_{n \to \infty} c_k(f^{(n)}) = c_k(f)$ for $k \geq 1$. From this, $\Gamma(f) = \Gamma$ and $\|f\|_{\infty} \leq A$. Since, on the other hand, $\|\Gamma\| \leq \|f\|_{\infty}$, we have $\|f\|_{\infty} = \|\Gamma\|$.

Corollary 2. For any function $F \in L_{\infty}$, there exists at least one function $h_0 \in H_{\infty}$ for which the distance $d(F, H_{\infty}) = \inf_{h \in H_{\infty}} \|F - h\|_{\infty}$ is attained.

Proof. Indeed, since $F \in L_{\infty}$, $\Gamma(F)$ is a bounded Hankel matrix. Consequently, there exists a minifunction $f_0 \in L_{\infty}$ such that $\Gamma(f_0) = \Gamma(F)$ and $\|f_0\|_{\infty} = \|\Gamma(F)\|$. Let $h_0 = F - f_0$; then $\Gamma(h_0) = 0$, which means $h_0 \in H_{\infty}$. Furthermore, $\|F - h_0\|_{\infty} = \|f_0\|_{\infty} = \|\Gamma(F)\|$. On the other hand, for any $h \in H_{\infty}$, we have $\|F - h\|_{\infty} \geq \|\Gamma(F - h)\| = \|\Gamma(F)\|$; therefore, $d(F, H_{\infty}) = \|\Gamma(F)\|$. This corollary could have been proven directly. Along our path, we obtained the following refinement:
$$d(F, H_{\infty}) = \|\Gamma(F)\| \quad (F \in L_{\infty}).$$ (1.12)
Up to this point, only the right-hand side of (1.3) has been used. Utilizing the left-hand side of (1.3) yields the following proposition:

Theorem (Generalization of Riesz's Theorem). Let $\{\gamma_j\}_1^{\infty}$ be a non-zero sequence of complex numbers. Then
$$1/\|\Gamma\| = \inf \{ \|g\|_1 : g \in H_1, (C, 1) \sum \gamma_m c_m(g) = 1 \}.$$ (1.13)
In this formulation, it is assumed that for an unbounded matrix $(\gamma_{j+k-1})_1^{\infty}$, $\|\Gamma\| = \infty$.

Proof. If $\|\Gamma\| < \infty$, then (1.3) holds. In determining the supremum of the left-hand side of (1.3), one can restrict the consideration to those $g \in H_1$ for which $\Phi_{\Gamma}(g) = 1$; then the left equality in (1.3) is equivalent to (1.13). If $\|\Gamma\| = \infty$, then by the theorem, the functional $\Phi_{\Gamma}(g)$ ($g \in H_1$) will be unbounded; consequently, $\sup \{ |\Phi_{\Gamma}(g)| : \|g\|_1 \leq 1 \} = \infty$. Restricting to those $g \in H_1$ for which $\Phi_{\Gamma}(g) = 1$, we obtain (1.13) for the case $\|\Gamma\| = \infty$.

§ 2. Hankel matrices

The reachable norm and the corresponding min-functions: the lower bound in $(1.13)$ can obviously be achieved only if $\|\Gamma\| < \infty$. Let $\|\Gamma\| < \infty$. A function $g \in H_1^k$ is called $\Phi_\Gamma$-extremal if $|\Phi_\Gamma(g)| = \|\Phi_\Gamma\| \cdot \|g\|_1 > 0$. Since $\|\Phi_\Gamma\| = \|\Gamma\|$, dividing a $\Phi_\Gamma$-extremal function by $\|\Phi_\Gamma\|$ yields a function that achieves the lower bound in $(1.13)$. Conversely, it is evident that any function $g \in H_1^k$ collinear to a function achieving the lower bound in $(1.13)$ is a $\Phi_\Gamma$-extremal function. A vector $\xi \in l_2$ is called $\Gamma$-extremal if $\|\Gamma\xi\| = \|\Gamma\| \cdot \|\xi\| > 0$. For such a vector, $|(\Gamma\xi, \eta)| = \|\Gamma\xi\| \cdot \|\eta\| = \|\Gamma\| \cdot \|\xi\| \cdot \|\eta\|$; consequently, $\Gamma\xi = s\eta$. These arguments are reversible; therefore, a vector will be $\Gamma$-extremal if and only if it is an eigenvector of the matrix $\Gamma^*\Gamma$ corresponding to the eigenvalue $\|\Gamma\|^2$ (and, consequently, $\|\Gamma\|$ is an $s$-number of the matrix $\Gamma$). In this (and only this) case, the pair of vectors $\{\xi, \eta\}$, where $\eta = \Gamma\xi / \|\Gamma\|$,

V. M. Adamyan, D. Z. Arov, and M. G. Krein will be an extremal pair for $\Gamma$, i.e.,
$$\eta = s^{-1}\Gamma\xi, \quad \xi = s^{-1}\Gamma^*\eta \quad (s = \|\Gamma\| \neq 0). \quad \text{(2.1)}$$

Theorem 2.1. Let $\Gamma = (\gamma_{j+k-1})_1^\infty$ be a bounded matrix. Then the lower bound in $(1.13)$ is achieved if and only if $\|\Gamma\|$ is an $s$-number of the matrix $\Gamma$. In this case, the following statements hold:
1) There exists a unique min-function $f_\mu$ corresponding to the matrix $\Gamma$.
2) The lower bound in $(1.13)$ is achieved for those and only those $g \in H_1^k$ (with $\Phi_\Gamma(g) = 1$) for which, almost everywhere,
$$f_\mu(\zeta) = \frac{g(\zeta)}{|g(\zeta)|} \cdot \|\Gamma\|. \quad \text{(2.2)}$$
3) Let $g$ be any function from the set described in statement 2, and let $g = b\phi$ be its canonical representation. Then almost everywhere,
$$f_\mu(\zeta) = \|\Gamma\| \cdot \frac{b(\zeta)\phi(\zeta)}{|\phi(\zeta)|}. \quad \text{(2.3)}$$
4) Under the conditions of statement 3, for any inner functions $b_1, b_2$ such that $b_1 b_2 = b$, an extremal pair $\{\xi, \eta\}$ for $\Gamma$ is defined by the equalities...
5) Let the vectors $\xi, \eta \in l_2$ constitute an extremal pair for $\Gamma$. Then there exist an outer function $\phi$ and an inner function $b$ such that the representation $(2.4)$ holds, and the function $\eta\chi$ will be $\Phi_\Gamma$-extremal, such that $f_\mu = \eta_- / \xi_+$.

As usual, we define $\text{sign } a = a/|a|$ for $a \neq 0$ and $\text{sign } a = 0$ for $a = 0$.

Proof. Let $f_\mu$ be any min-function for $\Gamma$ ($\Gamma(f_\mu) = \Gamma$, $\|f_\mu\|_\infty = \|\Gamma\|$). Then, according to $(1.9)$, for $g \in H_1^k$ with $\Phi_\Gamma(g) = 1$, we have:
$$1 = \Phi_\Gamma(g) = \left| \int f_\mu(\zeta) g(\zeta) d\zeta \right| \leq \int |f_\mu(\zeta)| |g(\zeta)| d\zeta \leq \|f_\mu\|_\infty \|g\|_1 = \|\Gamma\| \|g\|_1. \quad \text{(2.5)}$$
Starting from these relations, we will show that the existence of a function $g_\mu \in H_1^k$ on which the lower bound in $(1.13)$ is achieved implies statements 1)–4), and then we will prove statement 5). This will complete the proof of the theorem.
If $g_\mu \in H_1^k$ and $\Phi_\Gamma(g_\mu) = 1$ with $\|g_\mu\|_1 = 1/\|\Gamma\|$, then the equality sign must hold everywhere in the relations $(2.5)$ for $g_\mu$. Therefore, almost everywhere, $f_\mu(\zeta) g_\mu(\zeta) = |f_\mu(\zeta) g_\mu(\zeta)| = \|f_\mu\|_\infty |g_\mu(\zeta)|$, which leads to equality $(2.2)$. This proves statement 1).
Conversely, let $g \in H_1^k$, $\Phi_\Gamma(g) = 1$, and let $(2.2)$ hold. Then the equality sign will hold throughout $(2.5)$, and consequently, $\|g\|_1 = \|\Gamma\|^{-1}$. Thus, statement 2) is fully proved.
From the canonical representation $g = b\phi$, it follows that $\text{sign } g = \text{sign }(b\phi) = b\phi/|\phi|$. Thus, statement 3) follows from statement 2). Relation $(2.3)$ is equivalent to each of the following two: $f_\mu \bar{b}_1 \bar{\phi} = b_2 \phi$...

On Infinite Hankel Matrices: Writing the latter relations in terms of the Fourier coefficients of the functions $f_\mu$, and the functions $\eta_+$ and $\xi_-$ defined in $(2.4)$, we obtain

r i - S T i , fii^^sl (s=l|r||).

These relations are equivalent to relations (2.4). It remains for us to prove statement 5). Let $(\xi, \eta)$ be any extremal pair of unit vectors for $\Gamma$; that is, (2.1) holds and $\|\xi\| = \|\eta\| = 1$. As is easily seen, the equality $\Gamma \xi = s \eta$ can be written in terms of the functions $f, \xi_+, \eta_-$ as follows: $(f \xi_+ | \eta_-) = s$, where $\eta_-$ is the orthogonal projector of $L_2$ onto $H_2^\perp$. Therefore, for the function $(f \xi_+ | \eta_-)$, the equality of the extreme terms of this relation implies, according to (1.9), that $\Phi_f(\xi_+ \bar{\eta}_-) = 1$. On the other hand, we have $\|\xi_+ \eta_- \|_1 \le \|\xi_+ \|_2 \|\eta_- \|_2 = s^{-1} (= \|T\|^{-1})$. By virtue of (1.3), the inequality $<$ is impossible here, so $\|\xi_+ \eta_- \|_1 = \|T\|^{-1}$ and $\|\xi_+ \|_2 = \|\eta_- \|_2 = s^{-1/2}$. From the equality $\|\xi_+ \eta_- \|_1 = \|\xi_+ \|_2 \|\eta_- \|_2$ for the unit vectors $\xi, \eta$, it follows that $|\xi_+(t)| = |\eta_-(t)|$ almost everywhere. Therefore, if $\xi_+ = b_\xi \phi_\xi$ is the canonical representation for $\xi_+$, then the canonical representation for $\eta_-$ must include the same normalized outer function $\phi_\xi$, and thus the function $\eta_-$ admits the canonical representation $\eta_- = \bar{b}_\eta \bar{\phi}_\xi$. Thus, $\xi_+ \bar{\eta}_- = s^{-1} b_\xi b_\eta \phi_\xi^2 = s^{-1} b \phi_\xi^2$, and then $f = \|\Gamma\| \Phi_f / \xi_+ \bar{\eta}_- = \|\Gamma\| \bar{b} \bar{\phi}_\xi / \phi_\xi$. The theorem is proved.

Remark 2.1. From statements 3) and 5) of Theorem 2.1, the forward part of the following assertion follows: A bounded Hankel matrix has $\|\Gamma\|$ as its s-number if and only if it admits the representation:

$$\Gamma = \Gamma(s \bar{\phi}_e / b \phi_e) \quad (s > 0),$$

where $\phi_e$ is some outer function in $H_2$, and $b$ is some inner function; in this case $\|\Gamma\| = s$, and the multiplicity of $s$ as an s-number of the matrix is less than the degree of the function $b$ if $b$ is a rational function, and is equal to infinity otherwise. To this it can be added that the function $f_\Gamma = s \bar{\phi}_e / b \phi_e$ will be a minifunction. Indeed, if we form the functions $\xi_+$ and $\eta_-$ according to rule (2.4), we will again have (2.1), and hence $\|f_\Gamma\|_\infty = \|\Gamma\|$. The statement just formulated admits the following refinement (in the particular case of finite multiplicity of the s-number $\|\Gamma\|$).

Theorem 2.2. Let the bounded Hankel matrix $\Gamma$ have $\|\Gamma\|$ as its s-number of multiplicity exactly $n$. Then the corresponding (unique) minifunction $f_\Gamma$ admits the representation
$$f_\Gamma(t) = \|\Gamma\| \frac{\overline{\phi_e(t)}}{b(t) \phi_e(t)}, \quad \text{(2.6)}$$
the function $\phi_e$ in representation (2.6) is outer and is determined by $\text{sign } f_\Gamma$ (i.e., by $f_\Gamma$) up to a scalar real factor;

V. M. Adamyan, D. Z. Arov, M. G. Krein. The function $b$ specified in representation (2.6) is an inner function. If $\mathcal{P}_n$ is the set of all polynomials whose degree is less than $n$, then the formulas
$$\xi_+(t) = \frac{\phi_e(t) P(t)}{b(t)}, \quad \eta_-(t) = \frac{\overline{\phi_e(t) P(t)}}{t}, \quad \text{(2.7)}$$
where $P \in \mathcal{P}_n$, describe respectively all $\Gamma$-extremal vectors $\xi$ and all $\Gamma$-extremal vectors $\eta$.

Proof. By assumption, the equation $(\Gamma^* \Gamma - s^2 I) \xi = 0$, where $s = \|\Gamma\|$, has exactly $n$ linearly independent solutions. Therefore, there exists a solution $\xi^{(0)}$ of this equation whose first $n-1$ coordinates are zero. For a pair of extremal vectors $(\xi^{(0)}, \eta^{(0)} = \Gamma \xi^{(0)} / s)$, the representation $\xi_+^{(0)} = \phi_e b_1$ and $\eta_-^{(0)} = \bar{\phi}_e \bar{b}_2$ holds, where $b_1, b_2$ are inner functions and $\phi_e$ is an outer function from $H_2$. Consequently, the minifunction $f_\Gamma$ corresponding to the matrix $\Gamma$ factorizes in the form $f_\Gamma = s \bar{\phi}_e / b \phi_e$, where $b = b_1 b_2$ is an inner function. The function $b_1$ is a scalar (equal to one in modulus), for otherwise (see Remark 1.1) the multiplicity of $s^2$ as an eigenvalue of $\Gamma^* \Gamma$ would be greater than $n$. If we absorb $b_1$ into $\phi_e$, we obtain representation (2.6), where $\phi_e$ is an outer function. It follows from (2.6) that for the function $g = \xi_+ \bar{\eta}_- = \phi_e^2 / b$, the lower bound in (1.13) is achieved. Applying statements 3) and 4) of Theorem 2.1 to $g$, we find that the functions $\xi_k = t^k \phi_e / b$ ($0 \le k < n$) determine $\Gamma$-extremal vectors. Since these functions are linearly independent and their number is $n$, the first formula in (2.7) provides a description of all $\Gamma$-extremal vectors $\xi$. If $(\xi, \eta)$ is an arbitrary pair of extremal vectors for $\Gamma$ and $\xi_+ = P \phi_e / b$, then $\eta_- = s^{-1} \Gamma \xi_+ = s^{-1} P_- (f_\Gamma \xi_+) = P_- (\bar{\phi}_e P / b \cdot b) = \bar{\phi}_e \bar{P} / t$. This, taking into account statements 4) and 5) of Theorem 2.1, completes the proof of statement (2.7).

We now show that in representation (2.6), the function $\phi_e$ is determined up to a scalar real factor. Suppose that along with $\phi_e$, the function $\psi_e \in H_2$ is such that $f_\Gamma = s \bar{\psi}_e / b \psi_e$. Then the functions $t^k \psi_e / b$ ($k < n$) will determine $\Gamma$-extremal vectors, and therefore $t^k \psi_e / b = \phi_e P_k / b$ ($P_k \in \mathcal{P}_n$). But this is possible only if $P_k = c t^k$, i.e., $\psi_e = \rho \phi_e$, where $\rho$ is some scalar. Since $\text{sign } \psi_e^2 = \text{sign } \phi_e^2$, the scalar $\rho$ is real. The theorem is completely proved.

Remark 2.2. From the obtained description of $\Gamma$-extremal vectors, it is clear that in the case where $\Gamma$ has $\|\Gamma\|$ as its s-number of finite multiplicity $n$, one can specify a $\Gamma$-extremal vector $\xi$ with any pre-assigned first $n$ coordinates ($\xi_j = 0$ for $j \ge n$). It is also seen from (2.7) that in an extremal pair $(\xi, \eta)$ for $\Gamma$, we will have $\eta = \bar{\xi}$ if and only if the corresponding polynomial $P$ in (2.7) is symmetric, i.e., when $P(t) = t^{n-1} \overline{P(1/\bar{t})}$.

Remark 2.3. From the uniqueness of the definition of the function $\phi_e$ in representation (2.6), it follows that $\phi_e(\zeta)$ does not have zeros of order $\alpha \ge 1$ on the circle $|\zeta| = 1$; that is, there does not exist $\zeta_0$ with $|\zeta_0| = 1$ for which $\phi_e(\zeta) / (\zeta - \zeta_0) \in H_2$.

On infinite Hankel matrices. Indeed, otherwise for $\psi_e(\zeta) = \phi_e(\zeta) \frac{1 - \bar{\zeta}_0 \zeta}{\zeta - \zeta_0}$ we would have $\psi_e \in H_2$ and $\text{sign } \psi_e^2 = \text{sign } \phi_e^2$.

Remark 2.4. Using the results of Helson and Szegő [17], one can show that the following statement holds: In order for a bounded Hankel matrix to have $\|\Gamma\|$ as its isolated s-number of multiplicity $n$, it is necessary and sufficient that in representation (2.6) the function $\phi_e$ satisfies the condition
$$\ln |\phi_e(\zeta)| = u(\zeta) + \tilde{v}(\zeta), \quad \text{(2.10)}$$
where $u \in L_\infty$, and $\tilde{v}(\zeta)$ is the function harmonically conjugate to some function $v(t)$ for which $\|v\|_\infty < \pi/4$.

Remark 2.5. The condition of Theorem 2.2 is certainly satisfied if $\Gamma$ is a completely continuous matrix in $l_2$ and, moreover, a finite matrix. If $\Gamma$ is a finite matrix ($\gamma_j = 0, j > N$), then it follows directly from the statements of Theorems 2.1 and 2.2 that the outer function $\phi_e$ appearing in formula (2.6) is in fact a polynomial of degree not exceeding $N$ having no zeros in the disk $|z| < 1$, and that therefore $f_\Gamma$ is a rational function with its only singularity in the disk $|z| < 1$ being a pole of order at most $N$ at the point zero. The corresponding functions (2.8) are polynomials of degree no higher than $N$ and no lower than $n$. In this particular case, we obtain the well-known results of Carathéodory–Fejér [2] and F. Riesz [4], see also [11].

Remark 2.6. A matrix $\Gamma$ may correspond to an infinite set of essentially different minifunctions if $\|\Gamma\|$ is not its s-number. One should not think, however, that the condition that $\|\Gamma\|$ is an s-number of the matrix is necessary for the uniqueness of the minifunction. Consider, for example, the set of Hilbert matrices $\Gamma_\alpha = \|\frac{1}{j+k+\alpha}\|_{0}^\infty$, where $\alpha$ is any real number other than a negative integer. For $0 < \alpha < 1$, the spectrum of the matrix $\Gamma_\alpha$ consists of the segment $[0, \pi]$ and an isolated eigenvalue $\pi / |\sin \alpha \pi|$, and consequently, the matrix $\Gamma_\alpha$ corresponds to a unique minifunction $f_\alpha(\zeta)$, $f_\alpha(e^{i\theta}) = i e^{i\alpha(\theta-\pi)}$ ($0 < \theta < 2\pi$). For $\alpha > 1$, the spectrum of the matrix $\Gamma_\alpha$ consists of the segment $[0, \pi]$, where the number $\pi$ is not an eigenvalue of this matrix. However, for $1 < \alpha \le 3/2$, the matrix $\Gamma_\alpha$ corresponds to a unique minifunction $f_\alpha$, while for $\alpha > 3/2$ there is already an infinite set of minifunctions. A detailed study of Hilbert–Schur matrices and their corresponding minifunctions, based on the results of [13], [14], will be presented in another article.

*) In the memoir of F. Riesz [5], the question of the uniqueness of the solution to problem $B_0$ (see the introduction of the present article) was left open. A description of all solutions to problem $B_0$ was obtained by Ya. L. Geronimus [11].

V. M. Adamyan, D. Z. Arov, M. G. Krein

§ 3. Completely continuous Hankel matrices

and their corresponding functions. It is well known (see \cite{10}, p. 85) that for an arbitrary bounded operator $A$ in a Hilbert space:
$$\|A\|_\infty = s_\infty(A) \tag{3.1}$$
where $s_\infty(A)$ is the upper bound of the cluster spectrum of the non-negative operator $(A^*A)^{1/2}$, and $\mathfrak{S}_\infty$ denotes the ring of all completely continuous (compact) operators in the Hilbert space under consideration.

If we take $A$ to be a bounded Hankel matrix $\Gamma = (\gamma_{j+k-1})_1^\infty$ (and the operator in $l^2$ corresponding to this matrix), then relation (3.1) can be supplemented by the following theorem.

Theorem 3.1

For an arbitrary bounded Hankel matrix $\Gamma = (\gamma_{j+k-1})_1^\infty$:
$$s_\infty(\Gamma) = \min_{T_N} \|\Gamma - T_N\| = \inf_{T} \|\Gamma - T\| = \lim_{N \to \infty} \|\Gamma^{(N)}\| \tag{3.2}$$
where the infimum is taken over all finite-rank Hankel matrices $T$, and $\Gamma^{(N)}$ denotes the $N$-translated matrix $\Gamma^{(N)} = (\gamma_{j+k+N-1})_1^\infty$.

To prove this, we first establish a lemma.

Lemma 3.1

For a bounded Hankel matrix $\Gamma$ and any natural number $N$:
$$\min_{T_N} \|\Gamma - T_N\| = \|\Gamma^{(N)}\| \tag{3.3}$$
where the minimum is taken over all finite-rank Hankel matrices $T_N = (\tau_{j+k-1})_1^\infty$ for which $\tau_l = 0$ for $l > N$.

Proof. Let $\phi^{(N)}$ be the symbol (minifunction) for the matrix $\Gamma^{(N)} = (\gamma_{j+k+N-1})_1^\infty$. Then for the matrix $T_N = (\gamma_{j+k-1})_1^\infty - \Gamma(\zeta^N \phi^{(N)})$, we have $\tau_l = \gamma_l$ for $l > N$. Since $\zeta^N \phi^{(N)}$ is also a symbol, we have:
$$\|\Gamma - T_N\| = \|\Gamma(\zeta^N \phi^{(N)})\|_\infty = \|\Gamma^{(N)}\| \tag{3.4}$$
On the other hand, if $T_N = (\tau_{j+k-1})_1^\infty$ is an arbitrary finite-rank Hankel matrix ($\tau_l = 0$ for $l > N$), then $(\Gamma - T_N)^{(N)} = \Gamma^{(N)}$, and therefore:
$$\|\Gamma - T_N\| \geq \|(\Gamma - T_N)^{(N)}\| = \|\Gamma^{(N)}\| \tag{3.5}$$
Inequalities (3.4) and (3.5) yield the relation (3.3).

Proof of the Theorem. Since $\|\Gamma^{(N+1)}\| \leq \|\Gamma^{(N)}\|$ for $N = 1, 2, \dots$, there exists a limit:
$$s_\infty(\Gamma) = \lim_{N \to \infty} \|\Gamma^{(N)}\| \tag{3.6}$$

1^\1 J ^

11111 II1

$N \to \infty$ and by virtue of Lemma 3.1.

$$\inf \| \Gamma - G \| = \rho(\Gamma), \tag{3.7}$$
where the infimum is taken over all finite-rank Hankel matrices $G$. Since a finite-rank matrix corresponds to a finite-dimensional operator in $l_2$, it follows that
$$\|\Gamma - \Gamma\| \le \rho(\Gamma). \tag{3.8}$$

On Infinite Hankel Matrices

To complete the proof of the theorem, it remains to show that equality holds in (3.8). Suppose the contrary; that is, for some bounded Hankel matrix $\Gamma$, there exists a completely continuous matrix

$T = \{t_{j,k}\}_{j,k=0}^{\infty}$ such that

$$\|\Gamma - T\| \le \rho(\Gamma) - \alpha \quad (\alpha > 0).$$

Let $T^{(N)} = \{t_{j,k}\}_{j,k=N}^{\infty}$ ($N = 1, 2, \dots$). Then $\lim_{N \to \infty} \|T^{(N)}\| = 0$ because the matrix $T$ is completely continuous. We have
$$\rho(\Gamma) \le \|\Gamma^{(N)} - T^{(N)}\| + \|T^{(N)}\| \le \|\Gamma - T\| + \|T^{(N)}\| \le \rho(\Gamma) - \alpha + \|T^{(N)}\|.$$
Taking the limit as $N \to \infty$, we obtain $\rho(\Gamma) \le \rho(\Gamma) - \alpha$, which is impossible since $\alpha > 0$. The theorem is proved.

Corollary 3.1. A bounded Hankel matrix is completely continuous if and only if, for any $\epsilon > 0$, there exists a finite-rank Hankel matrix $\Gamma_\epsilon$ such that
$$\|\Gamma - \Gamma_\epsilon\| < \epsilon. \tag{3.9}$$

The main theorem of this section, in its expanded formulation, states:

Theorem 3.2. In order for a sequence of complex numbers $\{\gamma_j\}_1^\infty$ to be the sequence of "negative" coefficients of some continuous function $f(\zeta)$ ($|\zeta|=1$):

For the existence of a function $f \in C$ satisfying the conditions
$$f_j = \gamma_j \quad (j = 1, 2, \dots), \tag{3.10}$$
it is necessary and sufficient that the matrix $\Gamma = (\gamma_{j+k-1})_1^\infty$ be completely continuous (compact). When this condition is met, for any $\epsilon > 0$, there exists a continuous function $f$ satisfying conditions (3.10) such that $\|f\|_\infty < \|\Gamma\| + \epsilon$.

Proof. The necessity of the condition is well known (it was used implicitly in [14] and explicitly in [15]); that is, if $f \in C$, then the matrix $\Gamma(f)$ is completely continuous. Indeed, if $f \in C$, then by Fejér's theorem, $\|f - \Phi_N(f)\|_\infty \to 0$ as $N \to \infty$, where we define
$$\Phi_N(f) = \sum_{k=-N}^{N} \left(1 - \frac{|k|}{N+1}\right) c_k(f) \zeta^k. \tag{3.11}$$
For the trigonometric polynomial $g_N = \Phi_N(f)$, the matrix $\Gamma_N := \Gamma(g_N)$ is finite-rank. Since $\|\Gamma(f) - \Gamma_N\| \le \|f - \Phi_N(f)\|_\infty \to 0$ as $N \to \infty$, the matrix $\Gamma(f)$ is completely continuous as the limit of finite-rank operators.

We now prove the sufficiency of the condition. Suppose the matrix $\Gamma = (\gamma_{j+k-1})_1^\infty$ is completely continuous. Then, according to Corollary 3.1, for any $\epsilon > 0$, there exists a finite-rank Hankel matrix $\Gamma_1 = (\gamma_{j+k-1}^{(1)})$ such that $\|\Gamma - \Gamma_1\| < \epsilon/2^2$.

V. M. Adamyan, D. Z. Arov, M. G. Krein. Since the matrix $\Gamma - \Gamma_1$ is also completely continuous, there exists a finite-rank Hankel matrix $\Gamma_2 = (\gamma_{j+k-1}^{(2)})$ such that $\|\Gamma - \Gamma_1 - \Gamma_2\| < \epsilon/2^3$. Continuing this process by induction, we obtain a sequence of finite-rank Hankel matrices $\Gamma_n = (\gamma_{j+k-1}^{(n)})$ ($n = 1, 2, \dots$) such that

$$\left\| \Gamma - \sum_{j=1}^n \Gamma_j \right\| < \epsilon/2^{n+1}, \quad n \ge 1. \tag{3.12}$$

Let $\Gamma = \sum_{j=1}^\infty \Gamma_j$. Since the symbol of a finite-rank Hankel matrix (the "minifunction") is a rational function with no poles on the unit circle $|\zeta| = 1$, we have
$\Gamma_j = \Gamma(f_j)$ and $\|f_j\|_\infty = \|\Gamma_j\|$ ($j = 1, 2, \dots$). \tag{3.13}
We then have

$$\sum_{k=1}^n \|f_k\|_\infty < \|\Gamma\| + \epsilon$$

for $n \ge 1$.

l/„IU = ||r„|K

Consequently,

8 V--I

Thus, the series $\sum_{n=1}^{\infty} \|f_n\|$ converges and satisfies $\sum_{n=1}^{\infty} \|f_n\| < \|\Gamma\| + \epsilon$. Consequently, the series $\sum_{n=1}^{\infty} f_n$ converges uniformly to some function $f_{\epsilon} \in C$, where $\|f_{\epsilon}\| < \|\Gamma\| + \epsilon$. Based on equations (3.12) and (3.13), we obtain

$o(1)$ as ($n \to \infty$);

On the other hand, $|\Gamma(n) - \Gamma(f_{\epsilon})|$

$= o(1)$ as ($n \to \infty$),

from which it follows that $\Gamma = \Gamma(f_{\epsilon})$. Thus, we have constructed a function $f_{\epsilon} \in C$ such that $\Gamma = \Gamma(f_{\epsilon})$ and $\|f_{\epsilon}\| < \|\Gamma\| + \epsilon$. The theorem is proved.

Remark 3.1. In addition to Corollary 3.1, we have simultaneously obtained a simple rule for constructing approximating finite matrices $(\gamma_{j+k-1})_{1}^N$ for an arbitrarily given completely continuous Hankel matrix $\Gamma = (\gamma_{j+k-1})_1^{\infty}$. Specifically, if we set

$0$ for $j > N$,

then $\| \Gamma \| = \| \Gamma_d \|$. Indeed, if $f \in L^\infty$ is such that $\Gamma = \Gamma(f)$, then we have $\Gamma_d = \Gamma(\Phi_d(f))$.

Remark 3.2. The mini-function $f_\mu$ corresponding to a completely continuous Hankel matrix $\Gamma$ is, in general, not continuous.

Indeed, let $\xi(\zeta)$ be a real-valued continuous function such that its harmonic conjugate function $\chi(\zeta)$ is not continuous. Then for $\phi(\zeta) = \exp(i\chi(\zeta))$, we have $\phi, \phi^{-1} \in H^\infty$, and the function $f_\mu(\zeta) = \phi(\zeta) / \bar{\phi}(\zeta) = \exp\{2i\chi(\zeta)\}$ will be a mini-function. The matrix $\Gamma(f_\mu)$ can be represented in the form $\Gamma(f_\mu) = T_{\bar{\phi}} \Gamma(f_\mu |\phi|^2) T_{\bar{\phi}}$, where $T_{\bar{\phi}}$ is a bounded triangular Toeplitz matrix whose action is defined by the formula $(T_{\bar{\phi}} \xi)(\zeta) = \phi^{-1}(\zeta) \xi(\zeta)$ (for $\xi \in H^2$). Since $\phi(\zeta) \exp(2i\chi(\zeta)) \in C$, the matrix $\Gamma(\phi \exp(2i\chi))$ is completely continuous. Consequently, the matrix $\Gamma(f_\mu)$ is also completely continuous; at the same time, by construction, the mini-function $f_\mu(\zeta)$ is not necessarily continuous. As an example of a function $\chi \notin C$ for which $e^{i\chi} \notin C$, one may take (see \cite{9}, Ch. V, 2) the function $\chi = \sum n^{-2} \sin(2^n \theta)$ ($z = e^{i\theta}$).

In accordance with standard notation, let $A = H^\infty \cap C$. For functions $f \in C$, Corollary 1.2 can be supplemented by the following statement.

Theorem 3.4. If $f \in C$, then there exists a function $h \in A$ such that
$$ \| f - h \|_\infty = \min_{h \in A} \| f - h \|_\infty = \| \Gamma(f) \|. \quad (3.14) $$

Proof. If $f \in C$, then $\Gamma(f)$ is a completely continuous Hankel matrix. Therefore, for any $\epsilon > 0$, there exists a function $f_\epsilon \in C$ such that $\| \Gamma(f) - \Gamma(f_\epsilon) \| < \epsilon$. For $h_\epsilon = f - f_\epsilon$, we have $\Gamma(h_\epsilon) = 0$, which implies $h_\epsilon \in H^\infty \cap C = A$. Furthermore, $\| f - h_\epsilon \|_\infty = \| f_\epsilon \|_\infty < \| \Gamma(f) \| + \epsilon$. Consequently, $\inf_{h \in A} \| f - h \|_\infty \leq \| \Gamma(f) \|$. On the other hand, it is clear that $\inf_{h \in A} \| f - h \|_\infty \geq \min_{h \in H^\infty} \| f - h \|_\infty = \| \Gamma(f) \|$.

Remark 3.3.

Theorem 3.3 can also be obtained as a simple

This is a corollary of the Helson–Sarason theorem \cite{8}, which asserts that the linear manifold $C + H^\infty$ is closed in $L^\infty$. The approach we have chosen is distinguished by its constructive nature.

§ 4. Minifunctions from the Wiener ring

The theory of Hankel matrices $\Gamma = (\gamma_{j+k-1})_{1}^{\infty}$ and their corresponding mini-functions is significantly simplified when
$$ \sum_{j=1}^{\infty} |\gamma_j| < \infty. \tag{4.1} $$
In this case, it is directly verified that the transformation $\Gamma \xi = \eta$ is a completely continuous (compact) operator in each of the Banach spaces of vectors $l_1, l_2, l_{\infty}$ (where $l_{\infty}^0$ is the Banach space of sequences $\{\xi_j\}_1^{\infty}$ tending to zero, with the norm $\|\xi\| = \max |\xi_j|$). Consequently, the transformation $\Gamma^* \Gamma \xi$ will also be a completely continuous operator in each of these three spaces.

Lemma 4.1. Under condition (4.1), for any $\lambda \neq 0$, the equation
$$ \Gamma^* \Gamma \xi = \lambda^2 \xi \tag{4.2} $$
has the same solutions in all $l_k$ ($k = 1, 2, \dots, \infty$).

V. M. Adamyan, D. Z. Arov, M. G. Krein

Proof. Since the space $l_1$ is the conjugate of $l_{\infty}^0$, then by the well-known Riesz-Schauder theorem, equation (4.2) in $l_{\infty}^0$ and the conjugate equation $\Gamma \Gamma^* \eta = \lambda^2 \eta$ in $l_1$ have the same (finite) number of linearly independent solutions. On the other hand, if $\xi' \in l_1$ is a solution to the conjugate equation, then $\Gamma \xi' \in l_1$ will obviously be a solution to equation (4.2). Since $l_1 \subset l_k \subset l_{\infty}^0$ elementwise, Lemma 4.1 follows immediately. As usual, we shall denote by $W$ the Wiener algebra of continuous functions $f(\zeta) = \sum_{k=-\infty}^{\infty} \gamma_k \zeta^k$ such that $\sum |\gamma_k| < \infty$.

$$ \sum_{k=-\infty}^{\infty} |\gamma_k| < \infty $$

We denote its complex conjugate subalgebras by $W_+$ and $W_-$.

Theorem 4.1. For a function $f \in L_{\infty}$ ($\|f\|_{\infty} > 0$), the following statements are equivalent:
1) $f$ is a mini-function, and $e^{inf} f \in W_-$.
2) $f = \|f\|_{\infty} \psi / \overline{\psi} \cdot \zeta^{-n}$ for some integer $n \ge 0$, where $\psi \in W_+$ and the function $\psi$ is non-zero in the closed disk $|\zeta| \le 1$.
3) $f \in W$ and $\text{ind } f \le 0$.

When 1) or 2) is satisfied, the number $-\text{ind } f$ is equal to the multiplicity of the $s$-number $s = \|f\|_{\infty}$ of the matrix $\Gamma(f)$.

Proof. Let 1) hold. Then for $\Gamma = \Gamma(f)$, condition (4.1) is satisfied, and $\|\Gamma\| = \|f\|_{\infty}$ will be an $s$-number of some multiplicity for the matrix $\Gamma$. According to Theorem 2.2, $f = \|\Gamma\| \eta(\zeta) / \xi(\zeta)$, where $\xi, \eta$ are outer functions in $H_2$. If we define the vectors $\xi, \eta \in l_2$ by the equalities $\xi_k = \hat{\xi}_k, \eta_k = \hat{\eta}_k$, they will form an extremal pair of vectors for $\Gamma$, and consequently, the vector $\xi$ will be a solution to equation (4.2). By Lemma 4.1, we have $\xi \in l_1$, and thus $\xi(\zeta) \in W_+$, where $\xi(\zeta)$ does not vanish anywhere in the disk $|\zeta| < 1$. To obtain 2), it remains to show that $\xi(\zeta)$ does not vanish anywhere on the circle $|\zeta| = 1$. Suppose first that $s = \|\Gamma\|$. Then $\eta = \Gamma \xi / s$, so that $\Gamma \xi = s \eta$, i.e.,
$$ \sum_{k=1}^{\infty} \gamma_{j+k-1} \xi_k = s \eta_j \quad (j = 1, 2, \dots; s = \|\Gamma\|). \tag{4.3} $$
Assuming that for some $\alpha$ ($|\alpha| = 1$) the value $\xi(\alpha) = 0$, i.e., $\sum \xi_k \alpha^{k-1} = 0$...

form the vector

$$\Gamma_j \xi = \lambda^2 \xi_j + \lambda \eta_j + \rho_0 \eta_j \quad (j = 0, 1, \dots) \cdot (4.4)$$

For $\rho = 0$, we have

$\xi_k = \lambda \xi_{k-1} - \alpha \xi_k \quad (k = 1, 2, \dots, n, \xi_0 = 0)$.

For a function $f(\zeta)$ that is continuous and non-zero on the circle $|\zeta| = 1$, the symbol $\text{ind } f$ denotes the increment of its argument, divided by $2\pi$, as the point traverses the unit circle in the positive direction.

On Infinite Hankel Matrices

By performing the substitution (4.2), we obtain:
$$\eta_{j+k-1} = \lambda \eta_{j+k} + s (\eta_{j-1} + \eta_{j+k-1}) \quad (j = 1, 2, \dots), \tag{4.5}$$
$$-\alpha \eta_{j+k} = s \eta_{j-1} \quad (j = 1, 2, \dots), \tag{4.6}$$
$$\eta_{j+k-1} = s \eta_{j-1} \quad (j = 1, 2, \dots).$$

Substituting into (4.5) first $j+1$, then $j+2, \dots, j+m-1$ instead of $j$, and eliminating $\eta_{j+1}, \dots, \eta_{j+m-1}$ from the resulting equalities, we obtain:
$$\alpha^{m-1} \eta_{j+m} = (\lambda \eta_{j-1} - \alpha^{m-1} \eta_{j+m-1}). \tag{4.7}$$

Since according to (4.4) and (4.6) $\lim_{j \to \infty} \eta_j = 0$, by letting $m \to \infty$ in (4.7), we find that $\eta_j = -s \alpha \eta_{j-1}$ $(j = 1, 2, \dots)$. Recalling (4.5), we conclude that $\Gamma \eta' = s \eta'$, where $\eta' = \{\eta_{j-1}\}_1^\infty$. Obviously, $\eta' = s \xi'$, so that $\Gamma \eta' = s^2 \xi'$. Since $\xi' \in \ell^2$, on the basis of Lemma 4.1, $\xi' \in W_+$. It is easy to see that
$$\psi(z) = \Phi(z) / (z - \alpha) = (\Phi(z) - \Phi(\alpha)) / (z - \alpha) = \sum \xi_j z^j \quad (|z| < 1). \tag{4.8}$$

Thus, $\psi(\zeta) (\zeta - \alpha) = \Phi(\zeta)$ for $|\zeta| = 1$, where $\psi \in W_+$, and consequently, $f \psi / \Phi = -s \alpha \psi / \zeta^{n-1}$. Since $W_+ \subset H^2$, it follows from this last representation that $s = \|\Gamma\|$ is an $s$-number of the matrix $\Gamma$ of at least multiplicity two. We have reached a contradiction. Thus, for $n \ge 1$, it is proved that the function $\Phi(z)$ is non-zero in the closed disk $|z| \le 1$. This same assertion is valid for $n < 1$, as this case reduces to the previous one if, instead of the function $f$, we consider the function $f_1 = \zeta^{n-1} f$. Thus, 1) implies 2).

If $f \in W_+$, then $f^{-1} \in W_-$. If, in addition, the condition specified in 2) is satisfied for $W_+$, then by Wiener's theorem we also have $\phi^{-1} \in W_+$, and consequently, $f \in W$. Furthermore, in this case, according to the argument principle, $\text{ind } \phi = 0$, and therefore $\text{ind } f = -n$. Thus, 2) implies 3).

Now suppose 3) holds. Then one can define a continuous real function $g(\zeta)$ ($|\zeta|=1$) such that $\exp[i g(\zeta)] = \zeta^n f(\zeta) / \|f\|_\infty$ ($n = -\text{ind } f, |\zeta|=1$). By the well-known Wiener-Lévy theorem (see \cite{16}), $g \in W$ along with $f$. Let $g(\zeta) = g_+(\zeta) + g_-(\zeta)$ ($|\zeta|=1$); then $g_+ \in W_+$, $g_- \in W_-$, and $f(\zeta) = \|f\|_\infty \zeta^{-n} \phi_+ \phi_-$, where $\phi_+ = \exp(i g_+)$ and $\phi_- = \exp(i g_-)$. Since $g_+ \in W_+$, then $\phi_+ \in W_+$, and similarly $\phi_- \in W_-$, which means $f \in W$. It is also obvious that $\exp(i g_+(z))$ is an outer function in $A(D)$. Therefore, by Theorem 2.1, the function $f$ is a minifunction and, consequently, 3) implies 1). The theorem is proved, as its final assertion was obtained along the way.

Functional Analysis, Vol. 2, Issue 1

V. M. Adamyan, D. Z. Arov, M. G. Krein. As is well known \cite{16}, the Wiener–Lévy theorem admits broad generalizations. Accordingly, Theorem 4.1 can also be generalized to minifunctions from other Banach algebras. Let $\rho(n) = \rho(-n)$ ($n = 0, 1, 2, \dots$) be an even positive weight function satisfying the submultiplicative property: $\rho(n + m) \le \rho(n)\rho(m)$ ($n, m = 0, \pm 1, \dots$). Let $W_\rho$ be the corresponding Banach algebra,

$W_\rho = \{ f \in L^1 : \sum_{n=-\infty}^{\infty} |f_n| \rho(n) < \infty \}$, $W_\rho^+ = \{ f \in W_\rho : f_n = 0 \text{ for } n < 0 \}$, $W_\rho^- = \{ f \in W_\rho : f_n = 0 \text{ for } n > 0 \}$

be two complex-conjugate subalgebras. We set

$R = R_\rho = \lim_{n \to \infty} \sqrt[n]{\rho(n)}$.

It is known that $R \ge 1$. When $R > 1$, any function $f \in W_\rho$ admits an analytic continuation into the interior of the annulus $K_R = \{ z : R^{-1} < |z| < R \}$, which is continuous up to the boundary; when $R = 1$, we consider $K_R$ to coincide with the unit circle, and $W_\rho = W$. The "annulus" $K_R$ can be regarded as the set of maximal ideals of the Banach algebra $W_\rho$, and the homomorphism $f \to f(z)$ as the Gelfand homomorphism of the algebra $W_\rho$ into the algebra of functions on the set of maximal ideals (see \cite{16}). Therefore, according to the generalized Wiener–Lévy theorem \cite{16}, for a function $f \in W_\rho$ to admit the representation $f = \exp g$ with $g \in W_\rho$, it is necessary and sufficient that $\text{ind } f = 0$ and that the function $f$ does not vanish in the annulus $K_R$. If $R = 1$, the latter condition simplifies, as in this case $K_R$ is the unit circle.

Using this, it can be shown that for $R \ge 1$, Theorem 4.1 carries over completely to the case where the algebras $W$ and $W_\pm$ are replaced by the more general algebras $W_\rho$ and $W_\rho^\pm$. In particular, the theorem applies to the algebras $W_\kappa$ and $W_\kappa^\pm$ ($\kappa \ge 0$) generated by the weight function $\rho(n) = (1 + |n|)^\kappa$ ($\kappa \ge 0$). If $\kappa$ is a natural number, then the algebra $W_\kappa$ consists of precisely those functions $f$ that are $\kappa$ times continuously differentiable, with the $\kappa$-th derivative $f^{(\kappa)} \in W$. Therefore, one can assert, for example, that if the "principal part" $f_- = \pi_- f$ of a minifunction $f$ is $\kappa$ times continuously differentiable and its $\kappa$-th derivative is absolutely continuous with $f^{(\kappa+1)} \in L_2$, then the minifunction $f \in W_{\kappa+1}$, and consequently, it is at least $\kappa+1$ times continuously differentiable. Indeed, if a function $g(\zeta)$ is absolutely continuous and its derivative $g' \in L_2$, then $g \in W$.

If $R > 1$, then in any case it can be asserted that if the "principal part" $\pi_- f$ of a minifunction $f$ belongs to $W_\rho^-$, then the function $f$ is representable in the form $f = \phi / \psi$, where $\phi, \psi \in W_\rho^+$ and $\psi(z) \neq 0$ in the disk $|z| < R$. Therefore, the function $f$ will analytically continue to a function meromorphic in the annulus $K_R$, and all its poles will be located in some annulus $1 + \epsilon < |z| < R$, where $\epsilon > 0$ is sufficiently small. If for a minifunction $f$, its

On infinite Hankel matrices: principal part $\pi_- f$ is analytic on the unit circle, then the function $f$ itself is analytic on this circle. Indeed, if $\pi_- f$ extends to a holomorphic function outside the disk $|\zeta| \le \rho$ ($\rho < 1$), then for any $R < 1/\rho$, the function $\pi_- f \in W_\rho^-$, where $\rho(n) = R^{|n|}$ ($n = 0, \pm 1, \dots$).

Odessa Physical Institute of the State University
Odessa Pedagogical Institute named after K. D. Ushinsky
Odessa Civil Engineering Institute

Received by the Editorial Board
October 19, 1967

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