MATHEMATICS

V. P. ZAKHARYUTA

ON BASES IN THE SPACE OF FUNCTIONS ANALYTIC IN THE CLOSED DISK \(|z|\le R\)

(Presented by Academician A. N. Kolmogorov, 19 VII 1961)

In recent papers \((^{1,2})\) M. M. Dragilev proved a number of results for bases in \(A_R\)—the space of functions analytic in the disk \(|z|<R\), considered with the topology of uniform convergence on every closed subset of the disk \(|z|<R\). In \((^1)\) it is shown that every basis in \(A_R\) is a basis of regular convergence, i.e., if \(x(z)=\sum a_k x_k(z)\) is the basis expansion of an element, then

\[ \sum |a_k| \max_{|z|\le r} |x_k(z)|<\infty \]

for \(0\le r<R\). In \((^2)\) it is shown that every basis in \(A_R\), after suitable normalization and permutation of the elements, becomes the \(L\)-image of the power basis, i.e., is obtained from the power basis by applying a one-to-one and bicontinuous linear mapping of \(A_R\) onto itself.

In the present article analogous results are proved for \(\overline{A}_R\)—the space of functions analytic in the closed disk \(|z|\le R\), with the topology of the inductive limit

\[ \overline{A}_R=\lim_{s\to\infty}\operatorname{ind} A_{r_s} \qquad (r_s>r_{s+1}>R,\ r_s\to R). \]

In this topology a countable sequence \(x_n(z)\) converges to \(x(z)\) in \(\overline{A}_R\) if and only if there exists an \(s\) such that \(x_n,x\in A_{r_s}\) and \(x_n(z)\to x(z)\) in \(A_{r_s}\).

Up to isomorphism the relations \(\overline{A}_R^{\,*}=A_{1/R}\), \(A_{1/R}^{*}=\overline{A}_R\) hold exactly, i.e. \(\overline{A}_R\) is a reflexive space.

In what follows we shall need the following two propositions, proved by Newns \((^3)\).

\(1^\circ\). Let \(E\) be a complete countably normed space with topology \(T\). If every point \(x\in E\) is representable in the form of a series \(\sum f_k(x)x_k\), convergent in a topology \(T'\) weaker than the original topology \(T\), then \(\{f_k\}\) is a system of linear continuous functionals in \(E\), biorthogonal to \(\{x_k\}\).

\(2^\circ\). Let \(E=\lim_{s\to\infty}\operatorname{ind} E_s\), where \(E_s\) is a complete countably normed space and \(E_s\subset E_{s+1}\). Suppose that for some \(s\) every point \(x\in E_s\) can be represented by a series \(\sum a_k x_k\), convergent in the topology of the space \(E\). Then there exists \(\sigma(s)>s\) such that the series converges for every \(x\in E_s\) in the topology \(T_\sigma\) of the space \(E_\sigma\).

Lemma 1. Let \(x_k\) be a basis in \(E=\lim_{s\to\infty}\operatorname{ind} E_s\) and

\[ x=\sum f_k(x)x_k \tag{1} \]

representation of any element \(x\) in this basis. Then \(\{f_k\}\) is a system biorthogonal to \(\{x_k\}\).

It remains to show that the \(f_k\) are continuous functionals. By \(2^\circ\), for every \(x \in E_s\) the series (1) converges in the topology \(T_\sigma\) of the space \(E_\sigma\). The imbedding operation of \(E_s\) into \(E_\sigma\) is continuous; therefore the topology \(T_\sigma\) is weaker than the topology \(T_s\). Applying \(1^\circ\), we obtain that the \(f_k\) are continuous functionals in \(E_s\). Since \(s\) is arbitrary, this means that the \(f_k\) are continuous functionals in \(E\).

Lemma 2. Every basis in \(\overline A_R\) is a system biorthogonal to some basis in \(A_{1/R}\), and conversely, every system biorthogonal to some basis in \(A_{1/R}\) is a basis in \(\overline A_R\).

Let \(\{x_k\}\) be a basis in \(\overline A_R\); then, by Lemma 1,
\[ x=\sum f_k(x)x_k, \]
where \(f_k \in \overline A_R^{\,*}=A_{1/R}\). For every \(f \in A_{1/R}\) there is an expansion
\[ f(x)=\sum f(x_k)f_k(x), \]
where the series converges for each \(x \in \overline A_R\). By the reflexivity of \(\overline A_R\), the two kinds of weak convergence of functionals in \(\overline A_R\) coincide. Moreover, weak and strong convergence in \(A_{1/R}\) coincide. Therefore \(\{x_k\}\) is a basis in \(A_{1/R}\). The converse assertion is proved analogously.

We shall say that a basis in a linear topological space \(E\) is an unconditional basis if, after any permutation of its elements, it remains a basis in \(E\).

Theorem 1. Every basis in \(\overline A_R\) is an unconditional basis.

Let \(\{x_k\}\) be a basis in \(\overline A_R\); then, by Lemma 2, the biorthogonal system \(\{f_k\}\) is a basis in \(A_{1/R}\). But \(\{f_k\}\) is a basis of regular convergence in \(A_{1/R}\), and, moreover, an unconditional basis. Therefore \(\{f_{k_j}\}\) is a basis in \(A_{1/R}\) for every permutation \(k_j\) of the natural numbers. But then, by Lemma 2, \(\{x_{k_j}\}\) is a basis in \(\overline A_R\), as the system biorthogonal to a basis. This completes the proof.

Theorem 2. Every basis \(\{x_k\}\) in \(\overline A_R\) is the \(L\)-image of the basis obtained from the power basis by a certain permutation of the elements and by multiplying them by certain numbers.

Take in \(A_{1/R}\) a basis \(\{f_k\}\) biorthogonal to \(\{x_k\}\). It is obtained from the power basis by means of the transformations indicated in the theorem. Let these transformations be, successively,
\[ f_k' = z^{n_k},\qquad f_k''=t_k f_k',\qquad f_k=M f_k'', \]
where \(M\) is a one-to-one and bicontinuous linear mapping of \(A_{1/R}\) onto itself. The power basis \(\{z^n\}\) in \(A_{1/R}\) is biorthogonal to the same power basis \(\{z^n\}\) in \(\overline A_R\). Now carry out over \(\{z^n\}\) in \(\overline A_R\) successively the following transformations:
\[ x_k'=z^{n_k},\qquad x_k''=\frac{1}{t_k}x_k',\qquad x_k^{(0)}=(M^{-1})^*x_k''. \]

Obviously, \(\{x_k'\}\) and \(\{x_k''\}\) are biorthogonal respectively to \(\{f_k'\}\) and \(\{f_k''\}\). Then \(\{x_k^{(0)}\}\) is biorthogonal to \(f_k\). Indeed,
\[ (f_k,x_j^{(0)})=(f_k,(M^{-1})^*x_j'')=(M^{-1}f_k,x_j'')=(f_k,x_j'')=\delta_{k,j}. \]
By the uniqueness of the biorthogonal system, the basis \(\{x_k\}\) coincides with the basis \(\{x_k^{(0)}\}\). The theorem is proved.

We shall call a basis \(\{x_k\}\) in \(\overline A_R\) a basis of regular convergence if for every \(x \in \overline A_R\) there exists \(r(x)>1\) such that
\[ \sum |f_k(x)|\max_{|z|\le r(x)} |x_k(z)|<\infty . \]

Theorem 3. Every basis in $\bar A_R$ is a basis of regular convergence. It suffices to prove the theorem for bases that are $L$-images of the power basis. Let $x_k=M(z^k)$ be such a basis. Then, by (5), in view of the continuity of $M$, for every $r>R$ there exist $\rho(r)>R$ and $C_r>0$ such that

\[ \max_{|z|\le \rho(r)} |x_k(z)| \le C_r \max_{|z|\le r}|z^k| = C_r r^k \qquad (k=0,1,\ldots). \]

Now let $x(z)$ be an arbitrary element of $\bar A_R$, and let $\sum a_k z^k$ be the power-series expansion of the element $y=M^{-1}x$; moreover, obviously, $\sum |a_k|r^k<\infty$ for $r<r_0$, where $r_0$ is the radius of convergence of the power series for $y(z)$. On the other hand, $x=\sum a_k x_k$. By (2) we obtain

\[ \sum |a_k| \max_{|z|\le \rho(r)} |x_k(z)| \le C_r \sum |a_k|r^k<\infty . \]

This last inequality completes the proof of the theorem.

In conclusion, I consider it my pleasant duty to express my gratitude to my scientific adviser M. G. Khaplanov.

Rostov-on-Don
State University

Received
19 VII 1961

References

1. M. M. Dragilev, Nauchn. dokl. vyssh. shkoly, No. 4, 27 (1958).
2. M. M. Dragilev, UMN, 15, no. 2 (92), 181 (1960).
3. W. F. Newns, Phil. Trans. Roy. Soc. London, A245, No. 900, 429 (1953).
4. M. G. Khaplanov, DAN, 79, No. 6, 930 (1951).
5. K. M. Fishman, DAN, 127, No. 1, 40 (1959).