UDC 513.881

MATHEMATICS

S. Ya. KHAVINSON

ON THE RUDIN–CARLESON THEOREM

(Presented by Academician I. N. Vekua on 12 IV 1965)

Rudin \((^1)\) and Carleson \((^2)\) proved the following remarkable theorem (see also \((^3)\)). If \(F\) is a closed subset of points of the circle \(\Gamma: |\xi|=1\), having measure zero, and \(f(\xi)\) is an arbitrary continuous function on \(F\), then there exists a function \(\varphi(z)\), analytic in the disk \(|z|<1\), continuous in the closed disk \(|z|\leq 1\), and coinciding on \(F\) with \(f(\xi)\). Moreover, one may assume that \(\max |\varphi(z)|\), for \(|z|\leq 1\), coincides with \(\max |f(\xi)|,\ \xi\in F\). An extension of this theorem to the case of functional algebras more general than the algebra of continuous analytic functions was given by Bishop \((^4)\). A profound strengthening of this theorem was obtained by Pelczyński \((^{5,6})\), who also considered the space of continuous functions. We also point to a paper of Gamelin \((^7)\). In the present note we consider an analogue of the Rudin–Carleson theorem in the case of subspaces of an arbitrary Banach space \(X\).

Theorem 1. Let \(X\) be a Banach space, \(X^*\) its conjugate space, and suppose that the following conditions are fulfilled:

1. \(X^* = X_1^* \oplus X_2^*\), and the subspace \(X_2^*\) is weakly closed.

2. If \(x_1^*\in X_1^*\) and \(x_2^*\in X_2^*\), then \(\|x_1^*+x_2^*\|\geq \|x_2^*\|\).

3. A closed subspace \(Z\subset X\) is such that, if \(x^*(z)=0,\ z\in Z\), then \(x^*\in X_1^*\).

Denote by \(X_2\) the annihilator of \(X_2^*\) in the space \(X\): \(X_2=(X_2^*)_\perp\), and let \(Z_2=Z\cap X_2\). Then \(X/X_2\simeq Z/Z_2\) \((\simeq\) is an isomorphic isometry).

Remark. Let us first of all explain why this theorem is a generalization of the Rudin–Carleson result. Let \(X=C(\Gamma)\), the space of continuous complex-valued functions on \(\Gamma\); let \(Z\) be the subspace of \(C(\Gamma)\) consisting of functions admitting analytic continuation into the disk \(|z|<1\). The space \(X^*\) consists, as is known, of all finite Borel measures on \(\Gamma\). Let \(X_2^*\) consist of all those measures whose support is contained in \(F\), and \(X_1^*\) of the measures concentrated on \(\Gamma\setminus F\). Condition 1 is here, obviously, fulfilled. Condition 2 is fulfilled in exactly the same way, since in our case
\[ \|x_1^*+x_2^*\|=\|x_1^*\|+\|x_2^*\|. \]
If some functional \(x^*\) is orthogonal to \(Z\), then the measure defining it, by the F. and M. Riesz theorem, is absolutely continuous, and therefore its restriction to \(F\) is equal to zero, which means that condition 3 is fulfilled.

The subspace \(X_2\) is composed of all functions equal to zero on \(F\), and, as is easily seen, \(X/X_2\simeq C(F)\). The main assertion of the Rudin–Carleson theorem consists precisely in the fact that \(Z/Z_2\simeq C(F)\), and therefore follows from the theorem we have stated.

Proof of the theorem. For brevity denote \(R=X/X_2\) and \(R_1=Z/Z_2\).

Define a mapping of \(R_1\) into \(R\) in the following way. To the class \(\alpha=\{z_0+z_2\}\in R_1\), where \(z_0\) is a fixed element of \(Z\), and \(z_2\) runs through \(Z_2\), we put in correspondence the class \(\tilde{\alpha}=\{z_0+x_2\}\), where \(x_2\) runs through \(X_2\). The image of \(R_1\) under our mapping will be denoted by \(\widetilde{R}_1\). The mapping of \(R_1\) onto \(\widetilde{R}_1\) is one-to-one. Indeed, if \(\alpha_1\ne\alpha_2\), where \(\alpha_1=\{z_0'+z_2\}\) and \(\alpha_2=\{z_0''+z_2\}\), then \(z_0'-z_0''\notin Z_2\), and therefore \(z_0'-z_0''\notin X_2\), and consequently—

indeed, \(\tilde\alpha_1\ne \tilde\alpha_2\). It is now clear that the mapping \(R_1\) onto \(\widetilde R_1\) is an isomorphism. Our next step will be to prove that \(\widetilde R_1\) is dense in \(R\). The annihilator \(X_2^\perp\) of the subspace \(X_2\) in the space \(X^*\) coincides with \(X_2^*\) by the weak closedness of \(X_2^*: X_2^\perp=[(X_2^*)_\perp]^\perp=X_2^*\) (see, for example, \((^8)\), Ch. II, § 9). By the Hahn—Banach theorem this allows us to identify \(R^*\) with \(X_2^*\). Indeed, any functional \(\varphi\in R^*\) may be regarded as defined on \(X\) and orthogonal to \(X_2\); conversely, any \(x_2^*\in X_2^*\) may be regarded as a functional on \(R\), and it is also easy to see that if \(\varphi\in R^*\) and \(x_2^*\in X_2^*\) correspond to one another, then \(\|\varphi\|_R=\|x_2^*\|_X\). This follows from the fact that, on the one hand,
\(|x_2^*(x)|=|\varphi(\beta)|=|\varphi(\{x+x_2\})|\le \|\varphi\|_R\cdot\|\beta\|\le \|\varphi\|_R\cdot\|x\|\), and hence \(\|x_2^*\|_X\le \|\varphi\|_R\); and, on the other hand,

\[ \|\varphi\|_R=\sup_{\|\beta\|\le 1}|\varphi(\beta)| =\sup_{\|x+x_2\|\le 1}|x_2^*(x)| =\sup_{\|x+x_2\|\le 1}|x_2^*(x+x_2)| \le \sup_{\|t\|\le 1}|x_2^*(t)|=\|x_2^*\|_X . \]

Thus, \(R^*\simeq X_2^*\).

If now \(\varphi\in R^*\) and \(\varphi(\tilde\alpha)=0\) for \(\tilde\alpha\in\widetilde R_1\), then this means that the corresponding functional \(x_2^*\in X_2^*\) vanishes on \(Z\), and therefore, by assumption 3, \(x_2^*\equiv 0\). Hence \(\varphi\equiv 0\), and \(\widetilde R_1\) is dense in \(R\).

We shall now prove that \(R_1\simeq\widetilde R_1\). Since \(R_1=Z/Z_2\) is a Banach space, it will follow from the last isometry that \(\widetilde R_1\) is complete; and this, together with the density of \(\widetilde R_1\) in \(R\), leads to \(R_1=R\simeq R_1\). Consider the space \(Z^*\). By the Hahn—Banach theorem one may suppose that \(Z^*\) consists of the same functionals \(x^*\in X^*\), but now

\[ \|x^*\|_Z=\inf_{q^*\in Z^\perp}\|x^*-q^*\|\le \|x^*\|. \]

Here it is important to remember that condition 3 means the inclusion \(Z^\perp\subset X_1^*\). Condition 2 now gives, for every functional \(x_2^*\),

\[ \|x_2^*\|_Z=\inf_{q^*\in Z^\perp}\|x_2^*-q^*\|\ge \|x_2^*\|, \]

and consequently \(\|x_2^*\|_Z=\|x_2^*\|\). Thus, \(X_2^*\) may be regarded as isometrically embedded in \(Z^*\). Denote the image of \(X_2^*\) under this embedding by \(\widetilde X_2^*\). We shall show that \(\widetilde X_2^*\) is weakly closed in \(Z^*\). By the Krein—Šmulian theorem (see, for example, \((^9)\), p. 465) it suffices to show that the intersection of \(\widetilde X_2^*\) with an arbitrary closed ball \(S\) centered at the origin of \(Z^*\) is weakly closed. Let \(y^*\in Z^*\) be a limit point of the set \(\widetilde X_2^*\cap S\). This means that some generalized sequence \(\{\tilde x_2^{*l}\}\) (\(l\) ranges over the set of indices defining the generalized sequence) of elements of \(\widetilde X_2^*\cap S\) has \(y^*\) as its limit in the weak topology of the space \(Z^*\). Let \(T\) be the closed ball in \(X^*\) of the same radius as \(S\). The set \(X_2^*\cap T\) is bicompact in the weak topology of \(X^*\), and therefore the generalized sequence \(\{x_2^{*l}\}\), contained in \(X_2^*\cap T\) and corresponding to the sequence \(\{\tilde x_2^{*l}\}\), has in \(X_2^*\cap T\) a generalized limit point \(x_2^*\). From the definition of the weak topologies in \(Z^*\) and \(X^*\) it follows that on \(Z\) the functionals \(y^*\) and \(x_2^*\) coincide, and the latter means that \(y^*\in\widetilde X_2^*\). The closedness of \(\widetilde X_2^*\) in the weak topology of \(Z^*\) is proved. Evidently, we have \((\widetilde X_2^*)_\perp=Z_2\) and \(Z_2^\perp=\widetilde X_2^*\).

Now, for the coset \(\alpha=\{z_0+z_2\}\in R_1\) we obtain (again with the help of the Hahn—Banach theorem)

\[ \|\alpha\|=\inf_{z_2\in Z_2}\|z_0+z_2\| =\sup_{\substack{x_2^*\in \widetilde X_2^*\\ \|x_2^*\|\le 1}}|\tilde x_2^*(z_0)| =\sup_{\substack{x_2^*\in X_2^*\\ \|x_2^*\|\le 1}}|x_2^*(z_0)|. \]

For the coset \(\tilde\alpha=\{z_0+x_2\}\in\widetilde R_1\), corresponding to \(\alpha\) under the isomorphism of \(R_1\) onto \(\widetilde R_1\), we have

\[ \|\tilde\alpha\|=\inf_{x_2\in X_2}\|z_0+x_2\| =\sup_{\substack{x_2^*\in X_2^*\\ \|x_2^*\|\le 1}}|x_2^*(z_0)|=\|\alpha\|. \]

The isometry \(R_1\simeq\widetilde R_1\) is proved, and this completes the entire proof of the theorem.

Returning to the Rudin–Carleson theorem, let us note that from our theorem it follows only that the analytic function \(\varphi(z)\), extending \(f(\zeta)\), can be chosen so that
\[ \max_{|z|\leqslant 1} |\varphi(z)| \leqslant \max_{\zeta\in F} |f(\zeta)|+\varepsilon, \]
where \(\varepsilon>0\) is arbitrary. The possibility of extending \(f(\zeta)\) without increasing the norm does not follow directly from our theorem. In our terms, the possibility of such an extension would mean that for every \(z\in Z\) there is an element \(z_2^0\in Z_2\) such that
\[ \|z+z_2^0\|=\inf \|z+z_2\|,\quad z_2\in Z_2. \]

If \(D\) is a closed subset of \(\Gamma\setminus F\), then the function \(f(\zeta)\) on \(F\) can be extended to \(\Gamma\) continuously in such a way that the extending function is equal to zero on \(D\). If, however, one restricts oneself only to functions \(\varphi(z)\) analytic for \(|z|<1\), then for every \(\varepsilon>0\) there is an extension \(\varphi(z)\) such that \(|\varphi(z)|<\varepsilon\), \(z\in D\). These facts carry over to the abstract situation considered in the following way.

Theorem 2. Suppose that, under the hypotheses of Theorem 1, the subspace \(Q\subset X_1^*\) is closed in the weak topology of \(X^*\) and \(P=Q^\perp\). Then the image of \(P\) under the natural homomorphism \(X\to X/X_2\) coincides with all of \(X/X_2\). In other words, for every \(x\in X\) there exist \(x_2\in X_2\) and \(p\in P\) such that \(x=x_2+p\). Moreover, for every \(z\in Z\) and \(\varepsilon>0\) there exists \(z_2\in Z_2\) such that
\[ \sup_{q\in Q,\ \|q\|\leqslant 1} |q(z+z_2)| = \inf_{p\in P}\|z+z_2+p\|<\varepsilon. \]

Proof. Let \(U\) be the operator mapping \(P\) into \(X/X_2=R\) under the natural homomorphism \(X\to X/X_2\). To prove the first assertion of the theorem it is necessary, for the adjoint operator \(U^*\), to establish the inequality
\[ \|U^*(x_2^*)\|\geqslant m\|x_2^*\| \]
for some \(m>0\), since the space \(R^*=X_2^*\). But
\[ U^*(x_2^*)=x_2^*+q \]
with some \(q\in P^\perp=Q\). Since \(Q\subset X_1^*\), by assumption 2 of Theorem 1,
\[ \|x_2^*+q\|\geqslant \|x_2^*\|. \]

To prove the second assertion of the theorem, we shall show that
\[ \inf_{z_2\in Z_2,\ p\in P}\|z+z_2+p\| = \left\|\sup_{\|x^*\|\leqslant 1,\ x^*\in Z_2^\perp\cap Q}|x^*(z)|\right\| =0. \]

Every functional \(x^*\in Z_2^\perp\) has the form \(x^*=x_2^*+z^*\), where \(x_2^*\in X_2^*\), and \(z^*\in Z^\perp\subset X_1^*\). Since \(x^*\in Q\), we have \(x_2^*+z^*=q\in Q\subset X_1^*\). Hence it follows that \(x_2^*=0\), and this completes the proof.

Moscow Civil Engineering Institute
named after V. V. Kuibyshev

Received
5 IV 1965

REFERENCES

1. W. Rudin, Proc. Am. Math. Soc., 7, 808 (1956).
2. L. Carleson, Math. Zs., 66, 447 (1957).
3. K. Hoffman, Banach Spaces of Analytic Functions, IL, 1963.
4. E. Bishop, Proc. Am. Math. Soc., 13, 140 (1962).
5. A. Pelczynski, Stud. Math., 24, 285 (1964).
6. A. Pelczynski, Stud. Math., 25, 157 (1964).
7. T. W. Gamelin, Trans. Am. Math. Soc., 112, 278 (1964).
8. J. Loomis, An Introduction to Abstract Harmonic Analysis, IL, 1956.
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