UDC 517.948.3:517.948.42

MATHEMATICS

R. E. VAL’SKII

ON GLEASON PARTS FOR ALGEBRAS OF ANALYTIC FUNCTIONS AND MEASURES ORTHOGONAL TO THESE ALGEBRAS

(Presented by Academician V. I. Smirnov on December 6, 1965)

In the present note we consider a variant of the problem of uniform approximation of functions continuous on a set and analytic at its interior points by rational fractions. We assume that for separate parts of the original set such an approximation is possible, and seek conditions on the character of the “gluing” of these parts under which approximation is possible on the whole set. It turns out that approximation “as a whole” will be possible if the separate parts do not adjoin one another too closely.

The main part of the proof consists in decomposing measures defined on the original set and orthogonal to rational functions into a sum of measures concentrated on its subsets. This decomposition is stronger than is needed for obtaining the approximation theorem mentioned above, and is of independent interest. A similar decomposition was considered earlier in the works of Bishop \((^{1,2})\); for general Dirichlet algebras such a decomposition was constructed in the work of Glicksberg and Wermer \((^{3})\). The results obtained are applied to the study of one interesting class of sets.

In the note the following notation will be used: \(X\) is a compact set in the complex plane; \(\partial X\) is the boundary of \(X\); \(U\) is the interior of \(X\); \(U_1,\ldots,U_n,\ldots\) are the components of \(U\); \(C(X)\) is the algebra of all continuous functions on \(X\) with the usual norm; \(R(X)\) is the subalgebra of \(C(X)\) consisting of functions representable as the limit of a sequence of rational fractions with poles outside \(X\), converging uniformly on \(X\); \(A(X)\) is the algebra of functions continuous on \(X\) and analytic at the interior points of \(X\); \(R^\perp(X)\) is the space of all finite complex-valued Borel measures \(\mu\), defined on \(\partial X\) and orthogonal to \(R(X)\) (i.e. \(\int_{\partial X} f\,d\mu=0\), \(f\in R(X)\)); \(\|\mu\|=\int_{\partial X}|d\mu|\). The (Cauchy) potential \(F_\mu\) of a measure \(\mu\in R^\perp(X)\) is defined as follows:
\[ F_\mu(x)=\frac{1}{2\pi i}\int_{\partial X}\frac{d\mu(\xi)}{\xi-x}\quad (x\in U). \]
\(K(z,\delta)=\{w:\ |w-z|<\delta\}\); \(\overline{E}\) is the closure of the set \(E\). By the symbol \(f|_{E_0}\) we shall denote the restriction to the set \(E_0\) of a function \(f\) defined on the set \(E\), \(E_0\subset E\); \(\perp\) is the sign of mutual singularity of measures (two measures are called mutually singular if each is concentrated on a set of measure zero with respect to the other).

Let \(x_1,x_2\in X\); we shall write \(x_1\sim x_2\) if
\[ \|x_1-x_2\|_X=\sup_{\substack{f\in R(X)\\ \|f\|\le 1}} |f(x_1)-f(x_2)|<2. \]
This relation was introduced by Gleason \((^4)\) for arbitrary function algebras. Note that always \(\|x_1-x_2\|_X\le 2\). It can be shown \((^4)\) that the relation \(\sim\) is an equivalence relation (i.e. reflexive, symmetric, and transitive). An equivalence class in the sense of \(\sim\) is called a part. Each component \(U_i\) is contained entirely in one of the parts \((^4)\).

Let us pose the following question: can every measure \(\mu\in R^\perp(X)\) be decomposed into a sum of mutually singular measures \(\mu_i\in R^\perp(\overline U_i)\)?

Theorem 1. Let \(X\) be a plane compact set; let \(U_1,\ldots,U_n,\ldots\) be the components of the interior of \(X\); and let \(R(\partial X)=C(\partial X)\). In order that every measure \(\mu\in R^\perp(X)\) can be decomposed into a series convergent in variation
\[ \mu=\sum_i \mu_i, \]
where 1) \(\mu_i\) is concentrated on \(\partial U_i\), \(\mu_i\in R^\perp(\overline U_i)\), 2) \(\mu_i\perp\mu_j\) for \(i\ne j\), it is necessary and sufficient that \(U_i\) and \(U_j\) not belong to the same part when \(i\ne j\).

From general theorems of functional analysis and Theorem 1 there immediately follows

Theorem 2. Let \(X\) be a plane compact set; let \(U_1,\ldots,U_n,\ldots\) be the components of the interior of \(X\); suppose \(U_i\), \(U_j\) do not belong to one part for \(i\ne j\); and \(R(\partial X)=C(\partial X)\). If \(f\in C(X)\), \(f|_{\overline U_i}\in R(\overline U_i)\), \(i=1,\ldots\), then \(f\in R(X)\). In particular, if \(A(\overline U_i)=R(\overline U_i)\), \(i=1,\ldots\), then \(A(X)=R(X)\).

Proof of Theorem 1 (sufficiency). Let \(\mu\in R^\perp(X)\). Using the definition of a part, construct a sequence \(\{f_n:f_n\in R(X), \|f_n\|\le 1\}\) such that
\[ f_n(x)\to 1 \quad \left(x\in \bigcup_{j<i} U_j\right),\qquad f_n(x)\to 0 \quad \left(x\in \bigcup_{j\ge i} U_j\right). \]
Define the measure \(\nu_i\) as the weak limit of the measures \(f_n\mu\). Then \(\|\nu_i\|\le \|\mu\|\) and
\[ F_{\nu_i}(x)= \begin{cases} F_\mu(x), & x\in \displaystyle\bigcup_{j<i} U_j,\\ 0, & x\in \displaystyle\bigcup_{j\ge i} U_j. \end{cases} \]

Put \(\mu_i=\nu_{i+1}-\nu_i\). Then \(\mu_k\in R^\perp(\overline U_k)\). From comparison of Cauchy potentials it follows that
\[ \sum_k \mu_k=\mu. \]

Lemma 1. Let \(X_1,X_2\) be compact sets in the complex plane, \(R(X_1)=C(X_1)\). Let \(f\in C(X_1\cup X_2)\), \(f|_{X_2}\in R(X_2)\). Then \(f\in R(X_1\cup X_2)\).

Corollary. \(\mu_k\) is concentrated on \(\partial U_k\), \(\mu_k\in R^\perp(\overline U_k)\).

The mutual singularity of the measures \(\mu_i\) and \(\mu_j\) is established by

Lemma 2. Suppose \(U_i\) and \(U_j\) do not belong to the same part, \(\alpha\in R^\perp(\overline U_i)\), \(\beta\in R^\perp(\overline U_j)\). Then \(\alpha\perp\beta\).

Proof. From the definition of a part it follows that there exists a sequence \(\{g_n\}\) such that \(g_n\in R(X)\), \(\|g_n\|\le 1\), \(g_n(x)\to 1\) \((x\in U_i)\), \(g_n(x)\to -1\) \((x\in U_j)\). Then for any \(\alpha'\in R^\perp(\overline U_i)\), \(\beta'\in R^\perp(\overline U_j)\) we have
\[ \alpha'-\beta'=\lim_{n\to\infty} g_n(\alpha'+\beta'),\qquad \alpha'+\beta'=\lim_{n\to\infty} g_n(\alpha'-\beta') \]
(in the weak sense), whence \(\|\alpha'-\beta'\|=\|\alpha'+\beta'\|\). In particular,
\[ \|f\alpha-\beta\|=\|f\alpha+\beta\|\qquad (f\in R(\overline U_i)). \]
One can obtain such an equality for every \(f\in R(\partial X)=C(\partial X)\).

Let \(\alpha=h\beta+\gamma\) be the Radon–Nikodym decomposition of the measure \(\alpha\) with respect to the measure \(\beta\) (here \(h\in L'(\beta)\), \(\gamma\perp\beta\)). Choosing a bounded sequence \(\{f_n,\ f_n\in R(\partial X)\}\) such that \(f_nh\to |h|\) (with respect to the measure \(\beta\)), we obtain \(h=0\) (almost everywhere with respect to the measure), and this means precisely that \(\alpha\perp\beta\). In particular, \(\mu_i\perp\mu_j\). The theorem is proved.

We now consider the parts of compact sets of the following special form. Let \(l_1,l_2\) be rectifiable Jordan curves bounding domains \(U_1,U_2\), with \(U_1\cap U_2=\varnothing\), \(l_1\cap l_2=\varnothing\). Let \(X=\overline U_1\cup\overline U_2\). We shall say that a compact set \(X\) of this form belongs to the class \(K\).

The following theorem holds (by \(\operatorname{mes}_1\) we shall denote the natural measure on a rectifiable curve):

Theorem 3. 1) Let \(X\) be a compact set of class \(K\), bounded by the curves \(l_1,l_2\).

a) If \(\operatorname{mes}_1(l_1\cap l_2)>0\), then \(U_1\) and \(U_2\) lie in one part.

b) If \(\operatorname{mes}_1(l_1\cap l_2)=0\) and there exists a sufficiently smooth curve \(l\) such that \(U_1\) lies inside \(l\), and \(U_2\) outside it, then \(U_1\) and \(U_2\) lie in different parts.

2) There exists a compact set \(X_0\) of class \(K\) for which \(\operatorname{mes}_1(l_1\cap l_2)>0\), and almost all points of \(l_1\cap l_2\) belong to one part together with \(U_1\cup U_2\).

Remark. The part constructed in 2) has no analytic structure \((^4,^5)\).

Proof of 1a). Define the measures \(dz_1, dz_2\) as follows:
\(dz_i(E)=\displaystyle\int_{E\cap l_i} dz\) \((i=1,2;\ E\) is a Borel set). Since \(\operatorname{mes}_1(l_1\cap l_2)>0\), it follows that \(dz_1\) is not \(\perp dz_2\). But \(dz_i\in R^\perp(\overline{U_i})\) \((i=1,2)\), and from Lemma 2 it follows that \(U_1\) and \(U_2\) lie in one part.

1b) From the definition of equivalence it follows that if \(\widetilde X\supset X\) and \(U_1, U_2\) lie in different parts for \(\widetilde X\), then \(U_1\) and \(U_2\) lie in different parts for \(X\). Therefore one may assume that \(X\) is already bounded by sufficiently smooth curves.

Using estimates of analytic capacity on “good” curves \((^6,^7)\) and the theorem of S. N. Mergelyan \((^8)\), we obtain that \(A(X)=R(X)\). With the aid of Rudin’s theorem \((^9)\) one constructs a sequence \(\{f_n,\ f_n\in A(X)=R(X),\ \|f_n\|\le 1\}\) such that \(f_n(x)\to 1\) \((x\in U_1)\), \(f_n(x)\to -1\) \((x\in U_2)\). Our assertion now follows from the definition of equivalence.

One can approach this question in another way as well. Let \(\mu\in R^\perp(X)\). In \((^{11})\) it is shown that there exists a measure \(\mu'\), defined on \(l_1\), such that \(F_\mu=F_{\mu'}\) (in \(U_1\)). Then, for \(l_1\) sufficiently smooth, the potential \(F_\mu\) belongs to V. I. Smirnov’s class \(E_\delta\) in \(U_1\) for all \(\delta<1\) (for the definition of the class \(E_\delta\), see \((^{10})\)). Since \(F_\mu=0\) outside \(X\), it follows from the basic lemma of I. I. Privalov \((^{10})\) that \(F_\mu\) has almost everywhere on \(l_1\) summable angular boundary values (coinciding with the derivative, in the sense of Radon—Nikodym, of the measure \(\mu\) with respect to the measure \(dz_1\)). If now \(l_1\) belongs to the class \(C\), then, by the theorem of V. I. Smirnov \((^{10})\), \(F_\mu\in E_1\) in \(U_1\). In exactly the same way \(F_\mu\in E_1\) in \(U_2\). Bishop’s arguments \((^2)\) now show that \(\mu\in A^\perp(X)\). Hence \(A^\perp(X)=R^\perp(X)\) and \(A(X)=R(X)\).

2) Put
\[ X_0=\overline{K(0,1)}\setminus \bigcup_{i=1}^{\infty} K(z_i,r_i), \]
where \(\{z_i\}\) is a countable set dense on \([-1,1]\). If \(x_1,x_2\in X_0\), then
\[ \|x_1-x_2\|_{X_0}=\lim_{n\to\infty}\|x_1-x_2\|_{X_n}, \]
where
\[ X_n=\overline{K(0,1)}\setminus \bigcup_{i=1}^{n} K(z_i,r_i). \]
Moreover,
\[ \|x_1-x_2\|_{X_{n+1}}\le \|x_1-x_2\|_{X_n}+ \frac{4r_{i+1}}{\min(|x_1-z_{i+1}|,\ |x_2-z_{i+1}|)-r_{i+1}}. \]
If the \(r_i\) are chosen in accordance with this estimate, then \(X_0\) will have the required property.

In conclusion I express my gratitude to my scientific adviser V. P. Khavin for his help in this work.

Leningrad State University
named after A. A. Zhdanov

Received
25 X 1965

REFERENCES

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