Reports of the Academy of Sciences of the USSR
1968. Volume 180, No. 4

UDC 513.88+517

MATHEMATICS

V. P. Zakharyuta

ON THE QUASI-EQUIVALENCE OF BASES IN FINITE CENTERS OF HILBERT SCALES

(Presented by Academician L. V. Kantorovich on 24 VII 1967)

Let \(\{x_k\}\) and \(\{y_k\}\) be two bases in a linear topological space \(E\). The basis \(\{y_k\}\) is said to be quasi-equivalent to the basis \(\{x_k\}\) if there exist a permutation \(\{n_k\}\) of the natural series, a system of numbers \(\sigma_n\), and an isomorphism \(K\) of the space \(E\) onto itself such that

\[ x_k=\sigma_{n_k}K(y_{n_k}). \tag{1} \]

It is known \((^1)\) that in a Hilbert space all unconditional bases are quasi-equivalent to one another (and in (1) one may dispense with the permutation). On the other hand, the quasi-equivalence of all bases\(^*\) has been established for the so-called centers of nuclear Hilbert scales \((^2)\)\(^ {**}\) (in this case the permutation is already necessary). In the present paper the quasi-equivalence of unconditional bases in finite centers of completely continuous scales (not necessarily nuclear) is proved. Thus one of the problems posed in \((^2)\) is solved. The method proposed in the present paper not only makes it possible to dispense with the requirement that the space be nuclear, but also substantially simplifies the proof of quasi-equivalence (cf. \((^2), (^3)\)). The main theorem will be proved in Sec. \(4^\circ\).

\(1^\circ\). Let \(E\) be a complete countably Hilbert space and let \(\|x\|_p\), \(p=1,2,\ldots\), be a system of Hilbert norms defining the topology \(\mathfrak T\) of the space \(E\).

Under the indicated assumptions on the space \(E\), the following holds.

Lemma 1. Let \(\{x_k\}\) be an unconditional basis in \(E\) and let \(\{x'_k\}\) be the biorthogonal system in \(E^*\). Then the system of Hilbert norms

\[ \|x\|_{0,p}=\left(\sum |x'_k(x)|^2\|x_k\|_p^2\right)^{1/2} \]

defines on \(E\) the original topology \(\mathfrak T\).

This lemma is proved by applying Mazur’s result \((^4)\) that unconditional convergence of the series \(\sum x_k\) in a Hilbert space implies \(\sum \|x_k\|^2<\infty\), and by the usual arguments connected with the closed graph theorem.

\(2^\circ\). Using the concept of an \(n\)-diameter and its simplest properties \((^5, ^2)\), it is easy to prove the following lemma.

Lemma 2. Let \(H_0\subset H_1\subset H_2\subset H_3\) be a quadruple of Hilbert spaces with continuous embeddings; \(\{e_k\}\) a common orthonormal basis in \(H_0\) and \(H_3\); \(\{h_k\}\) a common orthonormal basis in \(H_1\) and \(H_2\); \(\|e_k\|_{H_0}=\|h_k\|_{H_1}=1\), \(\|e_k\|_{H_3}=\mu_k\downarrow 0\), \(\|h_k\|_{H_2}=\nu_k\downarrow 0\). Then there exists a constant \(C<\infty\) such that:

\[ \mu_k \le C\nu_k. \]

The assumption on the existence of bases \(\{e_k\}\) and \(\{h_k\}\) with the properties indicated in the lemma is equivalent to the assumptions that the embedding operator from \(H_1\) into \(H_2\) is completely continuous and that \(H_0\) is dense in \(H_3\).

* In a nuclear space all bases are absolute and, a fortiori, unconditional \((^2)\).

** The proof in \((^2)\) is carried out essentially by the method first applied in \((^3)\) in proving the quasi-equivalence of bases in the space \(A_r\).

3°. Let \(E_\alpha=E_\alpha(a_k)=\lim_{\lambda<\alpha}\operatorname{pr} H_\lambda\) be a finite (i.e. \(\alpha<\infty\)) center of the Hilbert scale \(H_\lambda=l_2(a_k^\lambda)\), \(-\infty<\lambda<\infty\); \(a_k\uparrow\infty\)—this condition ensures the complete continuity of the embedding \(H_\lambda\) into \(H_{\lambda_1}\), \(\lambda_1<\lambda\), i.e. the space \(E_\alpha\) will be Montel (6).*

Let \(\{e_k\}\) be a system of unit vectors in \(H_0=l_2\); then

\[ \|x\|_\lambda=\|x\|_{H_\lambda}=\left(\sum |(x,e_k)|^2 a_k^{2\lambda}\right)^{1/2}, \]

where \((x,y)=(x,y)_{l_2}\). Obviously, \(\{e_k\}\) is an unconditional basis in the space \(E_\alpha\); we shall call it the principal basis in \(E_\alpha\). Finite centers possess the following important property, which will be used essentially below:

Lemma 3. Let the operator \(T\in L(E_\alpha,H_{\lambda_0})\)** for some \(\lambda_0<\alpha\), and at the same time \(T\in L(H_\alpha,E_\alpha)\). Then \(T\in L(E_\alpha,E_\alpha)\).

This assertion is a consequence of the interpolation theorem for operators in Hilbert scales (7).

4°. We shall prove the main theorem.

Theorem. Let the space \(E_\alpha\) satisfy the assumptions of item 3° and let \(\{x_k\}\) be an unconditional basis in \(E_\alpha\). Then \(\{x_k\}\) is quasiequivalent to the principal basis \(\{e_k\}\) in \(E_\alpha\).

Proof. For simplicity we put \(\alpha=0\). Let \(\{x'_k\}\subset E_0^*\) be the system biorthogonal to the basis \(\{x_k\}\). By Lemma 1, the system of norms

\[ \|x\|_{0,\lambda}=\left(\sum |x'_k(x)|^2\|x_k\|_\lambda^2\right)^{1/2},\qquad \lambda<0, \]

is equivalent to the original system of norms \(\|x\|_\lambda\), \(\lambda<0\). Denote by \(H_\lambda^{(0)}\) the completion of the space \(E_0\) with respect to the norm \(\|x\|_{0,\lambda}\). From the equivalence of the systems of norms it follows that there exists \(\lambda_0<0\) such that, for \(\lambda:\lambda_0<\lambda<0\), numbers \(\mu_i=\mu_i(\lambda)<\infty\), \(i=1,2\), can be found which ensure the following continuous embeddings:

\[ H_{\mu_2}\subset H_\lambda^{(0)}\subset H_{\mu_1}. \tag{2} \]

Let now \(G_0\) be some Hilbert space for which:

a) the inclusions \(H_0\subset G_0\subset E_0\) hold with continuous embeddings;

b) \(\{x_k\}\) is an orthogonal basis in \(G_0\) (the existence of such a space \(G_0\) will be shown later).

Normalizing the system \(\{x_k\}\) in the space \(G_0\), we obtain the system \(\{\sigma_k x_k\}\), \(\sigma_k>0\), \(\|\sigma_kx_k\|_{G_0}=1\). Fix some \(\lambda:\lambda_0<\lambda<0\), and renumber the system \(\{\sigma_kx_k\}\) so as to obtain a system \(y_k=\sigma_{n_k}x_{n_k}\), ordered by decreasing norms in the space \(H_\lambda^{(0)}\): \(\|y_k\|_{0,\lambda}\downarrow0\). It will be shown below that the linear operator \(T\), defined by the relations

\[ Tx=\sum_{k=1}^{\infty}(x,e_k)_0 e_k, \tag{3} \]

is an isomorphism of the space \(E_0\) onto itself. The latter will mean the quasiequivalence of the bases \(\{x_k\}\) and \(\{e_k\}\), since \(e_k=\sigma_{n_k}K(x_{n_k})\), \(K=T^{-1}\).

First we establish the existence of a Hilbert space \(G_0\) with the properties a) and b) indicated above. Let \(\mathfrak M\) be the set of all nondecreasing functions \(M(\lambda)>0\), defined on \((-1,0)\). For each function \(M\in\mathfrak M\) consider the Hilbert space \(G(M)\) of all ele-

* The scale \(H_\lambda\) itself in this case is naturally called completely continuous.

** The notation \(T\in L(X,Y)\) means that either the operator \(T\) maps \(X\) linearly and continuously into \(Y\), or (if it is not defined on all of \(X\)) it admits a closure to a linear continuous operator from \(X\) to \(Y\).

ments from \(E\) having finite norm

\[ \|x\|_{G(M)}=\left\{\int_{-1}^{0}\|x\|_{0,\lambda}^{2}M(\lambda)^{2}\,d\lambda\right\}^{1/2}. \]

It is not difficult to see that for every \(M\in\mathfrak M\) the space \(G(M)\) is continuously embedded in \(E_0\) and the system \(\{x_k\}\) is an orthogonal basis in \(G(M)\). We shall show that one can choose \(M=M_0\in\mathfrak M\) so that \(H_0\) is continuously embedded in \(G(M_0)\). Indeed, the unit ball \(S\) of the space \(H_0\) is a bounded set in \(E_0\); therefore, for every \(\lambda<0\) there is \(C(\lambda)<\infty\) such that

\[ \|x\|_{0,\lambda}\le C(\lambda)<\infty,\qquad x\in S \]

(one may assume that \(C(\lambda)\) is increasing). Then \(S\subset U(M_0)\), where \(M_0(\lambda)=C(\lambda)^{-1}\in\mathfrak M\). The latter means that \(H_0\) is continuously embedded in \(G(M_0)\). The space \(G_0=G(M_0)\) thus constructed is the desired one.

Finally, let us show that the operator \(T\) (3) is an isomorphism of \(E_0\) onto itself. Since \(\{e_k\}\) and \(\{y_k\}\) are orthonormal bases, respectively, in \(H_0\) and \(G_0\), the operator \(T\) is an isomorphism of the space \(H_0\) onto \(G_0\). A fortiori,

\[ T\in L(H_0,E_0),\qquad T^{-1}\in L(H_0,E_0). \tag{4} \]

If we now establish that

\[ T\in L(E_0,H_{\mu_1}),\qquad T^{-1}\in L(E_0,H_{\mu_1}) \tag{5} \]

(for the definition of \(\mu_1\), see (2)), then from (4) and (5), by Lemma 3, we obtain that \(T\) is an isomorphism of the space \(E_0\) onto itself.

Thus it remains to prove the relations (5). Applying Lemma 2 to the quadruple of spaces \(H_0\subset G_0\subset H_\lambda^{(0)}\subset H_{\mu_1}\) and to the bases \(\{e_k\}\) and \(\{y_k\}\), we obtain

\[ \|e_k\|_{\mu_1}\le C\|y_k\|_{0,\lambda}. \tag{6} \]

From the same lemma, applied to the quadruple of spaces \(G_0\subset H_{-\varepsilon}\subset H_{\mu_2}\subset H_\lambda^{(0)}\), \(\varepsilon>0\), \(\mu_2+\varepsilon<0\), and to the bases \(\{y_k\}\) and \(\{\gamma_k e_k\}\), \(\gamma_k=a_k^\varepsilon\), it follows that

\[ \|y_k\|_{0,\lambda}\le C_1\|\gamma_k e_k\|_{\mu_2} = C_1\|e_k\|_{\mu_2+\varepsilon},\qquad C_1=C_1(\varepsilon)<\infty. \tag{7} \]

From (6) and (7) we obtain the inequalities

\[ \|Tx\|_{0,\lambda} =\left(\sum |(x,e_k)|^2\|y_k\|_{0,\lambda}^2\right)^{1/2} \le C_1\left(\sum |(x,e_k)|^2\|e_k\|_{\mu_2+\varepsilon}^2\right)^{1/2} = \]

\[ = C_1\left(\sum |(x,e_k)|^2 a_k^{2(\mu_2+\varepsilon)}\right)^{1/2} = C_1\|x\|_{\mu_2+\varepsilon}, \]

\[ \|T^{-1}y\|_{\mu_1} =\left(\sum |(x,e_k)|^2 a_k^{2\mu_1}\right)^{1/2} =\left(\sum |(x,e_k)|^2\|e_k\|_{\mu_1}^2\right)^{1/2} \le \]

\[ \le C\left(\sum |(x,e_k)|^2\|y_k\|_{0,\lambda}^2\right)^{1/2} = C\|y\|_{0,\lambda},\qquad x=T^{-1}y. \]

These inequalities mean, in view of (2), that \(T\in L(H_{\mu_2+\varepsilon},H_\lambda^{(0)})\subset L(E_0,H_{\mu_1})\), \(T^{-1}\in L(H_\lambda^{(0)},H_{\mu_1})\subset L(E_0,H_{\mu_1})\). The theorem is proved.

5°. Theorem 1 is equivalent to the following criterion for the isomorphism of the Köthe space \(L(b_{n,p})\) \((^2)\) to the finite center of a completely continuous Hilbert scale \(E_\alpha=E_\alpha(a_k)\).

Theorem 2. In order that the space \(L(b_{n,p})\) be isomorphic to the space \(E_\alpha\), it is necessary and sufficient that there exist a permutation \(\{n_k\}\) of the natural numbers and a sequence of numbers \(\sigma_k>0\) such that the spaces \(E_\alpha\) and \(L(C_{k,p})\), \(C_{k,p}=\sigma_k a_{n_k,p}\), coincide set-theoretically.

Rostov State University

Received
6 VII 1967

CITED LITERATURE

1. I. M. Gelfand, Uch. zap. MGU, 148, 4 (1951).
2. B. S. Mityagin, UMN, 16, no. 4 (1961).
3. M. M. Dragilev, UMN, 15, no. 2 (1960).
4. S. Mazur, Studia Math., 2 (1930).
5. A. Kolmogoroff, Ann. Math., 37, 107 (1936).
6. N. Bourbaki, Topological Vector Spaces, Moscow, 1959.
7. S. G. Krein, DAN, 130, no. 3 (1960).