B. E. VEITS

ON SOME STABILITY PROPERTIES OF BASES

(Presented by Academician V. I. Smirnov on 2 IV 1964)

Let \(E\) be a separable Banach space over the field of complex numbers. By the method of Shauder \((^{1})\), the unit ball \(S^*\) \((\|f\|\leqslant 1)\) of the space \(E^*\) can be turned into a metric compactum in which convergence of a sequence with respect to the metric is equivalent to its weak convergence.

We shall call the series \(\sum_{k=1}^{\infty} y_k\) regularly convergent if the series \(\sum_{k=1}^{\infty} |f(y_k)|\) converges uniformly with respect to \(f\in S^*\) (such series were called strongly unconditionally convergent by I. M. Gel'fand \((^{2})\)).

Lemma. The following four assertions are equivalent:

a) the series \(\sum_{k=1}^{\infty} y_k\) converges unconditionally;

b) the series \(\sum_{k=1}^{\infty} a_k y_k\) converges uniformly with respect to the choice of real numbers \(a_k\), \(a_k=\pm 1\), \(k=1,2,\ldots\);

c) the series \(\sum_{k=1}^{\infty} y_k\) converges regularly;

d) the series \(\sum_{k=1}^{\infty} \varepsilon_k y_k\) converges uniformly with respect to the choice of complex numbers \(\varepsilon_k\), \(|\varepsilon_k|\leqslant 1\), \(k=1,2,\ldots\).

Proof. The implication a) \(\Rightarrow\) b) follows from Kadec’s lemma \((^{3})\).

b) \(\Rightarrow\) c). Let \(f\in S^*\). Then for any \(x\in E\),
\(f(x)=f_1(x)+i f_2(x)\), where the functionals \(f_1\) and \(f_2\) are real, linear, and \(\|f_1\|\leqslant \|f\|\leqslant 1\), \(\|f_2\|\leqslant \|f\|\leqslant 1\).

Let \(|f_1(y_k)|=a_k f_1(y_k)\), \(|f_2(y_k)|=b_k f_2(y_k)\); \(a_k=\pm 1\), \(b_k=\pm 1\). Then for \(\varepsilon>0\), when \(n>m\geqslant N_\varepsilon\),

\[ \left\|\sum_{k=m}^{n} a_k y_k\right\|<\frac{\varepsilon}{2},\qquad \left\|\sum_{k=m}^{n} b_k y_k\right\|<\frac{\varepsilon}{2}, \]

\[ \sum_{k=m}^{n} |f(y_k)|\leqslant \sum_{k=m}^{n} |f_1(y_k)|+\sum_{k=m}^{n} |f_2(y_k)| = f_1\left(\sum_{k=m}^{n} a_k y_k\right)+ f_2\left(\sum_{k=m}^{n} b_k y_k\right)<\varepsilon. \]

c) \(\Rightarrow\) d). Let \(\varepsilon_k\), \(|\varepsilon_k|\leqslant 1\), be complex numbers. For natural \(n>m\geqslant N_\varepsilon\) there exists \(f_{mn}\in S^*\) such that

\[ \left\|\sum_{k=m}^{n} \varepsilon_k y_k\right\| = f_{mn}\left(\sum_{k=m}^{n} \varepsilon_k y_k\right) = \sum_{k=m}^{n} \varepsilon_k f_{mn}(y_k) \leqslant \sum_{k=m}^{n} |f_{mn}(y_k)|<\varepsilon \]

d) \(\Rightarrow\) a). If \((n_k)\) is an arbitrary increasing sequence of natural numbers, then, choosing \(\varepsilon_{n_k}=1\) and \(\varepsilon_n=0\) for \(n\ne n_k\), we obtain the convergence of the series
\[ \sum_{k=1}^{\infty} y_{n_k}=\sum_{n=1}^{\infty}\varepsilon_n y_n, \]
and this, by Orlicz’s theorem \((^4)\), also means a).

A system \((u_k)\) is called \(\omega\)-linearly independent if from the convergence of the series
\[ \sum_{k=1}^{\infty} c_k u_k=\theta \]
there follow the equalities \(c_k=0,\quad k=1,2,\ldots\)

In the papers \((^5,^6)\) (see also \((^7)\)) the following criterion for the stability of bases was established:

Theorem of M. G. Krein, M. A. Rutman, and D. P. Milman. If \((x_k)\) is a normalized basis of the space \(E\), \((f_k)\) is the system of functionals biorthogonal to \((x_k)\), and an \(\omega\)-linearly independent system \((u_k)\) satisfies the condition
\[ \sum_{k=1}^{\infty}\|u_k-x_k\|\cdot\|f_k\|<+\infty, \tag{1} \]
then the system \((u_k)\) is also a basis of the space \(E\), equivalent to the basis \((x_k)\).

We shall prove a stronger criterion for the stability of bases.

Theorem 1. If \((x_k)\) is a normalized basis of the space \(E\), and an \(\omega\)-linearly independent system \((u_k)\) satisfies the condition: the series
\[ \sum_{k=1}^{\infty}(u_k-x_k) \tag{2} \]
converges unconditionally, then the system \((u_k)\) is also a basis of the space \(E\), equivalent to the basis \((x_k)\).

Proof. By virtue of the lemma, the unconditional convergence of the series (2) implies the convergence of the series
\[ \sum_{k=1}^{\infty}|f(u_k-x_k)|, \tag{3} \]
uniformly with respect to \(f\in S^*\). By a lemma of I. M. Gelfand \((^8)\), from this it is easy to obtain the inequality
\[ \sum_{k=1}^{\infty}|f(u_k-x_k)|\le M\|f\|\le M. \tag{4} \]

If \((f_k)\), as above, is the system of functionals biorthogonal to \((x_k)\): \(f_i(x_k)=\delta_{ik}\), \(i,k=1,2,\ldots\), then, in view of the normalizedness of the basis \((x_k)\), \(\|f_k\|\le a,\ k=1,2,\ldots\)

For any \(m\) and \(n\) there is a functional \(\varphi_{mn}\in S^*\) such that
\[ \left\|\sum_{k=m}^{n} f_k(x)(u_k-x_k)\right\| = \sum_{k=m}^{n} f_k(x)\varphi_{mn}(u_k-x_k)\le \]
\[ \le \sum_{k=m}^{n}|f_k(x)|\,|\varphi_{mn}(u_k-x_k)| \le \left(\sum_{k=m}^{n}\|f_k\|\,|\varphi_{mn}(u_k-x_k)|\right)\|x\|\le \]
\[ \le a\sum_{k=m}^{n}|\varphi_{mn}(u_k-x_k)|\,\|x\| \le a\varepsilon\|x\|. \]

Consequently, the uniformly convergent series
\[ \sum_{k=1}^{\infty} f_k(x)(u_k-x_k)=Bx \]
defines a completely continuous operator \(B\). Since the equality
\[ Ax=(I+B)x=\sum_{k=1}^{\infty} f_k(x)u_k=\theta \]

is possible only (by virtue of the \(\omega\)-linear independence of the system \((u_k)\)) when \(f_k(x)=0,\ k=1,2,\ldots\), i.e., when \(x=\theta\). Hence the operator \(A\) is continuously invertible and, consequently, the system \((u_k)\), where \(u_k=(I+B)x_k=Ax_k,\ k=1,2,\ldots\), forms a basis in \(E\) equivalent to the basis \((x_k)\), as was required to prove.

Remark. By the theorem of Dvoretzky and Rogers \((^9)\), in the space \(E\) there exists a series \(\sum_{k=1}^{\infty} y_k\) which converges unconditionally, but not absolutely:

\[ \sum_{k=1}^{\infty} \|y_k\|=+\infty. \]

Put \(u_k=x_k-y_k\). Then, if \((x_k)\) is a normalized basis in the space \(E\), then, by Theorem 1, \((u_k)\) is a defective basis \((^{10})\) in \(E\), although the series \(\sum_{k=1}^{\infty}\|u_k-x_k\|\) diverges. Consequently, Theorem 1 is a strengthening of the theorem of M. G. Krein, M. A. Rutman, and D. P. Milman.

Following N. K. Bari \((^{11})\), we shall call a complete minimal system \((x_k)\subset E\) a Bessel system if, for every \(x\),

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

and a Hilbert system if, for any sequence \((\alpha_k)\in l^2\), there exists, and moreover uniquely, an element \(x\in E\) for which \(f_k(x)=\alpha_k,\ k=1,2,\ldots\), where \(f_i(x_k)=\delta_{ik},\ i,k=1,2,\ldots\). We shall call systems of elements \((x_k)\) and \((u_k)\) of the Banach space \(E\) weakly quadratically close if the series

\[ \sum_{k=1}^{\infty} |f(u_k-x_k)|^2 \tag{5} \]

converges uniformly with respect to \(f\in S^*\).

Theorem 2. If \((x_k)\) is a Bessel basis in the Banach space \(E\), then every \(l^2\)-linearly independent system of elements \((u_k)\subset E, *weakly quadratically close to the basis* \((x_k)\), is also a Bessel basis equivalent to the given basis \((x_k)\).

Corollary. If, in the space \(E\), the Bessel bases \((x_k)\) and \((u_k)\) are weakly quadratically close, then their biorthogonal systems are also weakly quadratically close.

Let now the space \(E=H\) be a separable Hilbert space. In the sequel, the main role is played by the following

Lemma. In order that a linear operator \(T\) be completely continuous, it is necessary and sufficient that, for every orthonormal basis \((e_k)\subset H\), the series

\[ \sum_{k=1}^{\infty} |(Te_k,x)|^2 \tag{6} \]

converge uniformly with respect to \(x\in H\) and \(\|x\|\le 1\).

If \(A\) is a linear continuous operator in \(H\), and \((e_k)\) is some orthonormal basis in \(H\), then in this basis the operator \(A\) is assigned the matrix \((a_{ik})_{1}^{\infty}\), \(a_{ik}=(Ae_k,e_i)\), \(i,k=1,2,\ldots\), and since

\[ |(Ae_k,x)|^2 = \left|\sum_{i=1}^{\infty} (x,e_i)(Ae_k,e_i)\right|^2 = \left|\sum_{i=1}^{\infty} a_{ik}(x,e_i)\right|^2, \]

* The system \((u_k)\) is called \(l^2\)-linearly independent if from the relations \(\sum c_k u_k=\theta\) and \(\sum |c_k|^2<+\infty\) it follows that \(c_k=0,\ k=1,2,\ldots\).

then a necessary and sufficient condition for the complete continuity of the operator \(A\) is the convergence, uniformly with respect to \(x \in H\) and \(\|x\| \leqslant 1\), of the series

\[ \sum_{k=1}^{\infty} \left| \sum_{i=1}^{\infty} a_{ik}(x,e_i) \right|^2 . \tag{7} \]

Condition (7) may be called the weak quadratability of the matrix \((a_{ik})_1^\infty\).

For brevity, let us call a basis weakly quadratically close to some orthonormal basis a \((W)\)-basis. Obviously, \((W)\)-bases are Riesz bases \({}^{(11)}\).

Theorems 3, 4, and 5 establish some properties and internal criteria for \((W)\)-bases.

Theorem 3. If \((x_k)\) is a \((W)\)-basis, then the biorthogonal system to it also forms a \((W)\)-basis, weakly quadratically close to the same orthonormal basis as the given basis \((x_k)\).

Theorem 4. In order that a Bessel basis \((x_k)\) be a \((W)\)-basis, it is necessary and sufficient that the biorthogonal systems \((x_k)\) and \((y_k)\), \((x_i,y_k)=\delta_{ik}\), \(i=1,2,\ldots\), be weakly quadratically close.

Corollary. In order that a Bessel basis \((x_k)\) be a \((W)\)-basis, it is necessary and sufficient that the operator \(I-C\) be completely continuous, where \(C\) is the operator carrying the biorthogonal system \((y_k)\) into the basis \((x_k)\):

\[ x_k = Cy_k,\quad k=1,2,\ldots \]

Theorem 5 (cf. \({}^{(12)}\)). In order that a complete system \((x_k)\) in \(H\) be a \((W)\)-basis, it is necessary and sufficient that: 1) the elements \((x_k)\), \(k=1,2,\ldots\), be \(l^2\)-linearly independent; 2) the matrix \(\bigl((x_i,x_k)-\delta_{ik}\bigr)_1^\infty\) be weakly quadratable.

Murmansk Pedagogical Institute

Received
19 III 1964

REFERENCES

\({}^{1}\) J. Schauder, Studia Math., 2 (1930).
\({}^{2}\) I. M. Gel'fand, Matem. sborn., 4 (46), 235 (1938).
\({}^{3}\) M. I. Kadets, UMN, 11, no. 5 (1956).
\({}^{4}\) W. Orlicz, Studia Math., 4, 27 (1933).
\({}^{5}\) M. G. Krein, M. A. Rutman, D. P. Mil'man, Zap. Kharkovsk. matem. obshch., (4), 16 (1940).
\({}^{6}\) M. G. Krein, L. A. Lyusternik, Mathematics in the USSR for 30 Years, 1948.
\({}^{7}\) B. E. Veits, Tr. Murmansk. vyssh. morekhodn. uchilishcha, 1, 1957.
\({}^{8}\) I. M. Gel'fand, Zap. Kharkovsk. matem. obshch., (4), 13 (1936).
\({}^{9}\) A. Dvoretzky, A. Rogers, Proc. Nat. Acad., 36, 192 (1950).
\({}^{10}\) I. Ts. Gokhberg, A. S. Markus, Izv. AN MoldSSR, no. 5, 103 (1962).
\({}^{11}\) N. K. Bari, Uch. zap. Mosk. univ., 4, no. 148 (1951).
\({}^{12}\) M. G. Krein, UMN, 12, no. 3 (1957).