Reports of the Academy of Sciences of the USSR

1. Volume 179, No. 3

MATHEMATICS

B. M. BYCHKOV, V. M. GROBER

ON THE ABSENCE OF AN UNCONDITIONAL BASIS

IN THE QUOTIENT SPACES \(L'_1/H_0^1\) AND \(L'/H'_0\)

(Presented by Academician A. N. Kolmogorov on 19 V 1967)

Up to now Banach’s problem has not been solved: does a basis exist in an arbitrary separable Banach space? As for unconditional bases, S. Karlin \((^2)\) was the first to prove that in the space \(C[0,1]\) of continuous functions on the segment \([0,1]\) there is no unconditional basis. The absence of an unconditional basis in the space \(L_1[0,1]\) of all real absolutely summable functions on the segment \([0,1]\) was shown by A. Pełczyński \((^3)\); he also posed the following question: does an unconditional basis exist in the quotient space \(L_1/H_0^1\)? The present paper gives a negative answer to this question. The absence of an unconditional basis in the quotient space \(L'/H'_0\) is also proved. In addition, it is shown that each of the indicated quotient spaces is not isomorphic to any subspace of a separable Banach space \(X^*\) conjugate to a Banach space \(X\), nor to any subspace of a Banach space with an unconditional basis.

We shall use the following notation: \(H^1\) is the Hardy space of functions \(f(z)\), analytic in the unit disk and such that

\[ \sup_{0<r<1}\int_0^{2\pi} |f(re^{i\theta})|\,d\theta=\|f\|<\infty,\qquad z=re^{i\theta}, \]

\[ H_0^1=\{f\in H^1:\ f(0)=0\}; \]

\(H'_0\) is the space of functions analytic in the unit disk, vanishing at the origin, and such that

\[ \int_0^1\int_0^{2\pi} |f(re^{i\theta})|\,r\,dr\,d\theta=\|f\|<\infty; \]

\(H_\infty\) is the \(B\)-space of analytic bounded functions in the unit disk with norm
\[ \|f\|_\infty=\operatorname*{ess\,sup}_{|z|=1}|f(z)|; \]
\(L'\) is the \(B\)-space of summable functions in \(|z|<1\).

It is known \((^5,\ \text{p. }195)\) that \((L_1/H_0^1)^*=H_\infty\). The space conjugate to \(L'/H'_0\) is, as is known \((^1)\), the annihilator of the space \(H'_0\), i.e. the set of those essentially bounded functions in the unit disk for which

\[ \iint_{|z|<1} z^n f(z)\,d\sigma=0,\qquad z=x+iy,\qquad d\sigma=dx\,dy,\qquad n=1,2,\ldots \]

Definition. A basis in a \(B\)-space is called unconditional if every system obtained by permuting its elements is also a basis.

Theorem 1. In the quotient space \(L_1/H_0^1\) there is no unconditional basis.

The proof rests on a number of lemmas.

Lemma 1. Let \(E\) be a separable subspace of \(H_\infty\). Then there exists a perfect set \(T \subset [0,2\pi]\) of positive measure such that every function \(\varphi \in E\) is equivalent to a function whose restriction to \(T\) is continuous.

Lemma 2. Let \(Z\) be a \(B\)-space satisfying one of the following conditions: a) \(Z\) is separable and is the conjugate of some \(B\)-space \((Z=X^*)\); b) \(Z\) contains an unconditional basis.

Then for every subspace \(Y \subset Z\) there exists a separable subspace \(E_Y \subset Y^*\) such that for every \(y_0^* \in Y^*\) there exists a sequence \(\{y_n^*\} \subset E_Y\) such that:

\[ \text{(I) }\quad y_0^*(y)=\lim_{n\to\infty} y_n^*(y)\quad \text{for every } y\in Y; \]

\[ \text{(II) }\quad \text{for every } y^{**}\in Y^{**} \text{ there exists } \lim_{n\to\infty} y^{**}(y_n^*). \]

Lemma 3. Let \(T \subset [0,2\pi]\) be a perfect set of positive measure. Then there exists a function \(\varphi_0(z)\in H_\infty\) whose restriction of angular boundary values to the set \(T\) is not equivalent to any function of the first Baire class.

Lemmas 1 and 2 were proved by Pelczyński \((^3)\); the proof of Lemma 3 will be given below.

For a measurable function put

\[ \Omega(\varphi,\Delta)=\sup_{t',\,t''\in\Delta}|\varphi(t')-\varphi(t'')|, \qquad \operatorname{ess}\Omega(\varphi,\Delta)=\inf_{\psi\sim\varphi}\Omega(\psi,\Delta), \]

\[ \operatorname{ess}\Omega(\varphi,t)=\lim_{\Delta_k\to t}\operatorname{ess}\Omega(\varphi,\Delta_k). \]

Let \(\{t_n=e^{i\theta_n}\}\) be a countable everywhere dense set in \(T\). Consider the following sets of functions:

\[ F=\{\varphi\in H_\infty:\|\varphi\|_\infty\leq 1\}, \]

\[ F_n=\{\varphi\in F:\operatorname{ess}\Omega(\varphi,t_n)\leq \alpha<1/10\}, \qquad n=0,1,\ldots, \]

\[ F^i_{n,k}=\{\varphi\in F:\operatorname{ess}\Omega(\varphi,\Delta_{n,k})\leq \alpha+1/i\}, \qquad i=5,6,\ldots,\ t_n\in\Delta_{n,k}. \]

Lemma \(3^1\). The set \(F\) with the metric

\[ \rho(\varphi_1,\varphi_2)=\frac{1}{2\pi}\int_0^{2\pi} \frac{|\varphi_1(e^{i\theta})-\varphi_2(e^{i\theta})|} {1+|\varphi_1(e^{i\theta})-\varphi_2(e^{i\theta})|} \,d\theta \]

is a complete metric space.

Lemma \(3^2\). The set \(F^i_{n,k}\) is closed.

Lemma \(3^3\). The set \(F^i_{n,k}\) is nowhere dense in \(F\).

In order to prove this, it is enough to verify that for every \(\varepsilon>0\) and every function \(f\in F^i_{n,k}\) one can find a function \(g\in F\), lying in the \(\varepsilon\)-neighborhood of \(f\), such that \(g\) does not belong to \(F^i_{n,k}\). Thus, Lemma \(3^3\) will be proved if the following is true.

Assertion 1. Let \(\Delta\) be some interval of \(|z|=1\), and let \(\beta>0\) be a fixed number. Then for every \(\varepsilon>0\) and every function \(f(z)\in F\) satisfying the condition \(\operatorname{ess}\Omega(f,\Delta)<\beta<1/3\), there exists a function \(g(z)\) such that

\[ g(z)\in F,\qquad \rho(f,g)<\varepsilon,\qquad \operatorname{ess}\Omega(g,\Delta)>\beta . \]

Proof. Suppose that \(\|f\|_\infty=1\) (the case \(\|f\|_\infty<1\) introduces nothing new). Consider the sequence of functions \(\{f_n(z)=f(r_n z)\}\), \(r_n<1\). Each of these functions is analytic in \(|z|\leq 1\).

As \(r_n \uparrow 1\), the sequence \(\{f_n(z)\}\) converges to the function \(f(z)\) in the norm \(H^1\) ((\(^{4}\), p. 89), and hence also in measure. Moreover,
\(\max_{|z|=r}|f(z)|\), strictly increasing, tends to one. Therefore, for the given \(\varepsilon>0\) (we shall assume \(\varepsilon<1/2\)) one can indicate a number \(N\) such that the inequalities
\[ \rho(f_N,f)<\varepsilon/4,\qquad 1-l_N=\delta<\varepsilon,\qquad \text{where } l_N=\|f_N\|_\infty<1. \]

Let us now make the following construction. With center at some interior point \(z_1\in\Delta\), take an arc \(\sigma\) so small that \(0<m\sigma<\delta/4\), and such that for any points \(z',z''\in\sigma\) (including also the endpoints of the interval \(\sigma\), which we denote by the points \(z_2,z_3\)) the condition
\[ |f_N(z')-f_N(z'')|<\delta/8 \]
is satisfied.

Let \(w_1'=f_N(z_1)\). Join the point \(w_1'\) to the origin; draw two circles
\(|w|=\delta/2\), \(|w|=\beta+3\delta/4\), and two rays symmetric with respect to \(Ow_1'\) at sufficiently small angles to it. The domain bounded by these rays, by the larger arc of the circle \(|w|=\delta/2\), and by the smaller arc of the circle \(|w|=\beta+3\delta/4\), will be denoted by \(G\), its boundary by \(\Gamma\), the points of intersection of the rays with the circle \(|w|=\delta/2\) by \(w_2,w_3\), and the point of intersection of the circle \(|w|=\beta+3\delta/4\) with the ray \(Ow_1'\) by \(w_1\).

Let \(\mu(z)\) be a function effecting a conformal mapping of \(|z|\leq 1\) onto the domain \(G\) with the additional conditions: the points \(z_i\) pass into the points \(w_i\) \((i=1,2,3)\). Such a function exists (by Riemann’s theorem), is analytic in \(|z|<1\), and continuous in \(|z|\leq 1\). It is then shown that the function \(g(z)=f_N(z)-\mu(z)\) is the required one.

Completion of the proof of Lemma 3. The inclusion
\[ F_n\subset \sum_{k=1,\, i=5}^{\infty} F_{n,k}^i \]
is valid. Since \(F_{n,k}^i\) is nowhere dense in \(F\), it follows that \(F_n\) is a set of the first category;
\[ \sum_{n=1}^{\infty} F_n \]
is also a set of the first category, and consequently the set
\[ F_0=F-\sum_{n=1}^{\infty} F_n \]
is nonempty. Suppose that \(\varphi_0\in F_0\). Then for it
\(\operatorname{ess}\Omega(\varphi_0,t_n)>\alpha,\ n=1,2,\ldots\), and since \(\{t_n\}\) is everywhere dense in \(T\), no point of the set \(T\) is a point of continuity of the function \(\varphi_0\) and of its equivalents. The lemma is proved.

The proof of Theorem 1 is now carried out according to a scheme analogous to Pelczyński’s scheme (\(^{3}\)).

Theorem 2. In the quotient space \(L'/H_0'\) there is no unconditional basis.

The general scheme of the proof of Theorem 2 is analogous to the scheme of the proof of Theorem 1. The difference will occur only in the proof of the main assertion.

Assertion 2. Let \(\Delta\) be some plane set of positive measure in \(|z|<1\); let \(\beta>0\) be a fixed number. Then for every \(\varepsilon>0\) and every function \(f\in F\) satisfying the condition
\[ \operatorname{ess}\Omega(f,\Delta)\leq \beta<1/10, \]
there exists a function \(g(z)\) such that
\[ g(z)\in F,\qquad \rho(f,g)<\varepsilon,\qquad \operatorname{ess}\Omega(g,\Delta)>\beta . \]

Proof. Among the points of the set \(\Delta\) there exists at least one point \(z_1\), in an arbitrarily small neighborhood of which there is a set of positive measure from \(\Delta\). For the number \(\varepsilon/2\) and the function \(f(z)\) construct a piecewise-constant function \(f_0(z)\in F\) close to \(f(z)\). Take, besides the point \(z_1\), also some point \(z_2\), and draw a circle through them.

Moreover, choose the point \(z_2\) so that the area of the disk \(K\) (bounded by the circle passing through \(z_1,z_2\)) is less than \(\varepsilon/2\).

Let \(w_1=f_0(z_1)\). Join the point \(w_1\) to the origin and draw from the origin two rays symmetric with respect to \(Ow_1\), making sufficiently small angles with it. We consider three cases: the 1st case \(|w_1|\geq 1/2\); the 2nd case \(|w_1|<1/2,\ |w_1|\ne0\); the 3rd case \(|w_1|=0\).

In the 1st case denote by \(D\) the domain bounded by the two indicated rays and by the smaller arcs of the circles \(|w|=|w_1|\) and \(|w|=|w_1|-1/4\); denote by \(w_2\) and \(w_3\) the points of intersection of the circle \(|w|=|w_1|-1/4\) with the rays.

In the 2nd case denote by \(D\) the domain bounded by the same rays and by the smaller arcs of the circles \(|w|=|w_1|\) and \(|w|=|w_1|+1/4\), and by \(w_2,w_3\) the points of intersection of the same rays with the circle \(|w|=|w_1|+1/4\).

In the 3rd case, when \(w_1=0\), the domain \(D\) will be the sector formed by the arc of the circle \(|w|=1/4\) enclosed between two arbitrary rays issuing from the origin, between which the angle is sufficiently small, while \(w_2,w_3\) are the points of intersection of the circle \(|w|=1/4\) with these rays.

Let \(\mu(z)\) be a function realizing a conformal mapping of the disk \(K\) onto the domain \(\overline D\) with the additional conditions: the points \(z_i\) go into the points \(w_i\) \((i=1,2)\). Such a function exists by the Riemann theorem, is analytic inside the disk \(K\), and is continuous in the closed disk.

Construct the function

\[ \nu(z)= \begin{cases} \mu(z), & z\in \overline K,\\ 0, & z\notin \overline K. \end{cases} \]

It is then proved that the function \(g(z)=f_0(z)-\nu(z)\) will be the desired one. This assertion is proved.

Remark. Analogously to Theorem 1, using part a) of Lemma 2 and part b) for \(Y\ne Z\), and also the separability of the space \(L_1/H_0^1\) \((L'/H_0')\), we obtain Theorem 3.

Theorem 3. The space \(L_1/H_0^1\) \((L'/H_0')\) is not isomorphic to any subspace of a separable Banach space \(X^*\) conjugate to a Banach space \(X\), nor to any subspace of a Banach space with an unconditional basis.

In conclusion the authors express their gratitude to A. Pelczynski for posing the problem.

References

1. M. M. Day, Normed Linear Spaces, IL, 1961.
2. S. Karlin, Bull. Am. Math. Soc., 54, 148 (1948).
3. A. Pelczynski, Colloquium math., 8, fasc. 2 (1961).
4. I. I. Privalov, Boundary Properties of Analytic Functions, 1950.
5. K. Hoffman, Banach Spaces of Analytic Functions, IL, 1963.