MATHEMATICS

B. E. VEITS

ON SOME CHARACTERISTIC PROPERTIES OF UNCONDITIONAL BASES

(Presented by Academician A. N. Kolmogorov, November 29, 1963)

1. Let \(E\) be a Banach space over the field of complex numbers. It is easy to prove that for unconditionally convergent series the following holds.

Proposition 1. If the series \(\sum_{k=1}^{\infty} x_k\) converges unconditionally, then the series

\[ \sum_{k=1}^{\infty} |f(x_k)| \]

converges uniformly with respect to \(f \in E^*\) and \(\|f\| \leqslant 1\).

Let us agree to denote by the symbol \([x_k]\) the closure of the linear span of the system of elements \((x_k) \subset E\).

Proposition 2. In order that a basis \((x_k)\) of the space \(E\) be unconditional, it is necessary and sufficient that, for every increasing sequence \((n_k)\) of natural numbers, the subspace \([x_{n_k}]\) have a complement in the space \(E\).

Proof. Necessity. Let \((f_k)\) be the system of functionals biorthogonal to \((x_k)\): \(f_i(x_k)=\delta_{ik}\), \(i,k=1,2,\ldots\). If \((n_k)\) is an increasing sequence of natural numbers, then, by Orlicz’s theorem \((^1)\), the series \(\sum_{k=1}^{\infty} f_{n_k}(x)x_{n_k}\) converges for every \(x \in E\), and, since \((x_{n_k})\) is a basis in \([x_{n_k}]\), this series converges to some \(y \in [x_{n_k}]\):

\[ \sum_{k=1}^{\infty} f_{n_k}(x)x_{n_k} = y = \sum_{k=1}^{\infty} \varphi_{n_k}(y)x_{n_k}, \]

where the functionals \(\varphi_{n_k}\), \(k=1,2,\ldots\), are the restrictions of the corresponding functionals \(f_{n_k}\) to \([x_{n_k}] \subset E\). Thus every element \(x \in E\) is represented, evidently uniquely, in the form of a sum:
\[ x=y+z;\qquad y \in [x_{n_k}];\qquad z=x-y \in [x_n]_{n\ne n_k};\qquad E=[x_{n_k}] \oplus [x_n]_{n\ne n_k}. \tag{1} \]

Sufficiency. If for an arbitrary increasing sequence \((n_k)\) of natural numbers (1) holds, then, obviously,
\(f_{n_k}(x)=f_{n_k}(y)\), and for every \(x \in E\) the series
\(\sum_{k=1}^{\infty} f_{n_k}(x)x_n\) converges, which, by the cited theorem of Orlicz, completes the proof.

Proposition 3. In order that a basis \((x_k)\) be unconditional in the space \(E\), it is necessary and sufficient that, for every \(x\), the series

\[ \sum_{k=1}^{\infty} |f_k(x)|\,|f(x_k)| \tag{2} \]

converge uniformly with respect to \(f \in E^*\), \(\|f\| \leqslant 1\).

The validity of this assertion follows from Proposition 1.

Corollary. If \((x_k)\) is an unconditional basis in the space \(E\), then for every \(x \in E\) the series
\[ \sum_{k=1}^{\infty} |f_k(x)|\,x_k \]
converges.

Remark. If \((x_k)\) is a basis in \(E\), then the convergence, for all \(x\), of the series
\[ \sum_{k=1}^{\infty} |f_k(x)|\,x_k \]
is not sufficient for the given basis to be unconditional. Indeed, consider the following example.

Example. Consider, in the space \(l\) of absolutely summable numerical sequences, the system of elements
\[ f_k=(0,\ldots,0,1,-1,0,\ldots),\qquad k=1,2,\ldots \]
where the first nonzero entry is in the \(k\)-th position. This system is biorthogonal to the system \((x_k)\) of elements of the space \(c_0\):
\[ x_k=(1,\ldots,1,0,\ldots),\qquad k=1,2,\ldots, \]
where the last \(1\) is in the \(k\)-th position, which forms a basis in \(c_0\). Hence the system \((f_k)\) forms a basis in \([f_k]\subset l\). It is easy to see that \([f_k]\ne l\). Therefore the basis \((x_k)\) in \(c_0\) and the basis \((f_k)\) in \([f_k]\) are not unconditional, since otherwise, taking into account the weak completeness of the space \(l\), by Karlin’s theorem \((^2)\), the system \((f_k)\) would have to be a basis in the space \(l\). Meanwhile, one can prove that if the series
\[ \sum_{k=1}^{\infty} a_k x_k \]
converges, then the series
\[ \sum_{k=1}^{\infty} |\alpha_k|\,x_k \]
also converges. The following, however, is true.

Theorem 1. For a basis \((x_k)\) to be unconditional, it is necessary and sufficient that for every \(x\in E\) the series
\[ \sum_{k=1}^{\infty} |f_k(x)|\,x_k \]
converge and that the inequality
\[ \left\|\sum_{k=1}^{\infty} f_k(x)\,x_k\right\| \le \left\|\sum_{k=1}^{\infty} |f_k(x)|\,x_k\right\|, \tag{3} \]
hold, where the constant \(M\) does not depend on the element \(x\in E\).

Proof. Necessity. If the basis \((x_k)\) is unconditional, then the corollary implies convergence of the series
\[ \sum_{k=1}^{\infty} |f_k(x)|\,x_k. \]
If
\[ \bar f_k(x)=\varepsilon_k |f_k(x)|;\qquad |\varepsilon_k|=1;\qquad k=1,2,\ldots, \]
then, applying the estimate of L. A. Gurevich \((^3)\), we obtain
\[ \left\|\sum_{k=1}^{\infty} f_k(x)\,x_k\right\| = \left\|\sum_{k=1}^{\infty} \varepsilon_k |f_k(x)|\,x_k\right\| \le M\left\|\sum_{k=1}^{\infty} |f_k(x)|\,x_k\right\|. \]

Sufficiency. Let \((x_k)\) be a basis, and suppose that for every \(x\) the series
\[ \sum_{k=1}^{\infty} |f_k(x)|\,x_k \]
converges and that estimate (3) holds. If \(x\in E\) and \(f\in E^*\), then by \((\varepsilon_k)\) denote a sequence of complex numbers such that \(|\varepsilon_k|=1\),
\[ \varepsilon_k f_k(x) f(x_k)=|f_k(x)|\,|f(x_k)|. \]
If \(\varepsilon>0\), then
\[ \left\|\sum_{k=p}^{q} |f_k(x)|\,x_k\right\| < \frac{\varepsilon}{M\|f\|} \quad\text{for } q>p\ge N_\varepsilon . \]

Denote by \(y\) the sum \(\sum_{k=p}^{q}\varepsilon_k f_k(x)x_k\). Then

\[ \sum_{k=p}^{q}|f_k(x)|\,|f(x_k)| =\sum_{k=p}^{q}\varepsilon_k f_k(x)f(x_k) =f\left(\sum_{k=p}^{q}\varepsilon_k f_k(x)x_k\right) =f\left(\sum_{k=1}^{\infty}f_k(y)x_k\right)\leq \]

\[ \leq \|f\|\left\|\sum_{k=1}^{\infty}f_k(y)x_k\right\| \leq M\|f\|\left\|\sum_{k=1}^{\infty}|f_k(y)|x_k\right\| =M\|f\|\left\|\sum_{k=p}^{q}|f_k(x)|x_k\right\|<\varepsilon \]

and, consequently, the basis \((x_k)\) is unconditional.

Theorem 2. In order that the basis \((x_k)\) be unconditional, it is necessary and sufficient that, for arbitrary \(n\) and \(x\), the inequalities

\[ m\left\|\sum_{k=1}^{n}|f_k(x)|x_k\right\| \leq \left\|\sum_{k=1}^{n}f_k(x)x_k\right\| \leq M\left\|\sum_{k=1}^{n}|f_k(x)|x_k\right\| \tag{4} \]

hold.

In the proof one uses Theorem 1 and an estimate of L. A. Gurevich \((^3)\).

1. A system \((u_k)\subset E\) will be called unconditionally \(\omega\)-linearly independent if, from the unconditional convergence of the series \(\sum_{k=1}^{\infty}c_k u_k=\theta\) to the zero element \(\theta\in E\), there follow the equalities \(c_k=0;\ k=1,2,\ldots\).

Definition. We shall say that a basis \((x_k)\) of a space \(E\) is \(\Phi\)-stable if every unconditionally \(\omega\)-linearly independent system \((u_k)\subset E\) for which the series

\[ \sum_{k=1}^{\infty}\|u_k-x_k\|f_k \tag{5} \]

converges is also a basis of the space \(E\).

Theorem 3. In order that a basis \((x_k)\subset E\) be unconditional, it is necessary and sufficient that it be \(\Phi\)-stable.

Proof. Necessity was proved by the author \((^4)\) in the case where the space \(E\) is reflexive. If one takes into account Proposition 3, the proof also goes through for an arbitrary space.

Sufficiency. Suppose that the basis \((x_k)\) is \(\Phi\)-stable, but not unconditional. Then there exist \(x_0\in E\) and \(f_0\in E^*\) such that the series \(\sum_{k=1}^{\infty}|f_k(x_0)|\,|f_0(x_k)|\) diverges. Let \(|f_k(x_0)|\,|f_0(x_k)|=\varepsilon_k f_k(x_0)f_0(x_k)\). Denote by \(\mathcal L\) the set of functionals that are linear combinations of the functionals \((f_k)\) with nonnegative coefficients:

\[ \mathcal L=\left\{f;\quad f=\sum_{k=1}^{s}\beta_k f_k;\ \beta_k\geq 0\right\}. \]

Let \(\varphi\in\mathcal L\) and \(\varphi(x_0)\neq0\). Obviously, \(\varphi(x_k)\geq0\). Consider the system

\[ y_k=\frac{1}{\varepsilon_k}x_k-\frac{x_0}{\varepsilon_k\varphi(x_0)}\varphi(x_k);\quad k=1,2,\ldots . \tag{6} \]

This system is unconditionally \(\omega\)-linearly independent. Indeed, let

\[ \sum_{k=1}^{\infty}c_k y_k=\theta \]

and let the series converge unconditionally. Then the series

\[ \sum_{k=1}^{\infty}|f_0(c_k y_k)| = \sum_{k=1}^{\infty}|c_k|\,|f_0(y_k)| \]

converges, and for the chosen \(\varepsilon_k\) the series

\[ \sum_{k=1}^{\infty}\varepsilon_k c_k f_0(y_k) = \sum_{k=1}^{\infty}c_k f_0(\varepsilon_k y_k) \]

converges absolutely.

From equality (6) it follows that the series

\[ \sum_{k=1}^{\infty}c_k f_0(x_k) = \sum_{k=1}^{\infty}c_k f_0(\varepsilon_k y_k) + \frac{f_0(x_0)}{\varphi(x_0)} \sum_{k=1}^{s}c_k\varphi(x_k). \]

Let

\[ \sum_{k=1}^{\infty}\frac{c_k}{\varepsilon_k}\varphi(x_k) = \sum_{k=1}^{s}\frac{c_k}{\varepsilon_k}\varphi(x_k) =\alpha. \]

Then

\[ \sum_{k=1}^{\infty}\frac{c_k}{\varepsilon_k}x_k = \frac{x_0}{\varphi(x_0)} \sum_{k=1}^{\infty}\frac{c_k}{\varepsilon_k}\varphi(x_k) = \frac{\alpha x_0}{\varphi(x_0)}; \tag{7} \]

\[ c_k=\varepsilon_k f_k\left(\frac{\alpha x_0}{\varphi(x_0)}\right) = \frac{\alpha}{\varphi(x_0)}\varepsilon_k f_k(x_0); \qquad k=1,2,\ldots; \tag{8} \]

\[ \sum_{k=1}^{\infty}c_k f_0(x_k) = \frac{\alpha}{\varphi(x_0)} \sum_{k=1}^{\infty}\varepsilon_k f_k(x_0)f_0(x_k) = \frac{\alpha}{\varphi(x_0)} \sum_{k=1}^{\infty}|f_k(x_0)|\,|f_0(x_k)|. \tag{9} \]

Equality (9) is possible only when \(\alpha=0\), and consequently, by (8), \(c_k=0\), \(k=1,2,\ldots\). Therefore the system \((y_k)\), and together with it the system \((\varepsilon_k y_k)\), is unconditionally \(\omega\)-linearly independent. And since the series

\[ \sum_{k=1}^{\infty}\|\varepsilon_k y_k-x_k\|f_k = \sum_{k=1}^{\infty}\frac{\|x_0\|}{|\varphi(x_0)|}\varphi(x_k)f_k = \frac{\|x_0\|}{|\varphi(x_0)|} \sum_{k=1}^{s}\varphi(x_k)f_k \]

converges, by the assumption of the theorem the system \((\varepsilon_k y_k)\) forms a basis of the space \(E\). The latter, however, is impossible, since

\[ \varphi(\varepsilon_k y_k) = \varphi(x_k)-\frac{\varphi(x_0)}{\varphi(x_0)}\varphi(x_k)=0; \qquad k=1,2,\ldots, \]

and, consequently, the system \((\varepsilon_k y_k)\) is not even complete.

Remark. The convergence condition for the series (5) is weaker than the condition of M. G. Krein, M. A. Rutman, and D. P. Milman \((^5)\).

Theorem 4. In order that a normalized basis \((x_k)\) in Hilbert space be a Riesz basis \((^6)\), it is necessary and sufficient that it be \(\Phi\)-stable.

The proof is based on Theorem 3 and on known theorems of N. K. Bari \((^6)\) and I. M. Gelfand \((^7)\).

Theorem 5. If \(\{x_k(t)\}\) is an unconditional basis in \(L^p_{[a,b]}\) with biorthogonal system of functions \(\{f_k(t)\}\subset L^q\) \((p^{-1}+q^{-1}=1)\), then every \(\omega\)-linearly independent system of functions \(\{u_k(t)\}\subset L^p\) for which

\[ \int_a^b \left[ \sum_{k=1}^{\infty} \|x_k-u_k\|_{(p)}^2\,|f_k(t)|^2 \right]^{q/2} dt<+\infty, \]

is also an unconditional basis in \(L^p\), equivalent to the given one.

Murmansk Pedagogical Institute

Received
29 XI 1963

References

1. W. Orlicz, Studia Math., 1 (1929).
2. S. Karlin, Duke Math. J., 15, 971 (1948).
3. L. A. Gurevich, UMN, 7, no. 5 (1953).
4. B. E. Veits, UMN, 17, no. 6 (1962).
5. M. G. Krein, M. A. Rutman, D. P. Milman, Zapiski Kharkov Mathematical Society, (4), 16 (1940).
6. N. K. Bari, Scientific Notes of Moscow University, 4, issue 148 (1951).
7. I. M. Gelfand, Scientific Notes of Moscow University, 4, issue 148 (1951).