UDC 513.88

MATHEMATICS

V. P. ZAKHARYUTA

ON THE QUASI-EQUIVALENCE OF BASES IN AN INFINITE CENTER OF A HILBERT SCALE

(Presented by Academician L. V. Kantorovich, March 28, 1969)

1°. Let \(\{x_k\}\) be an unconditional Schauder basis in a linear topological space \(E\). It is obvious that the system

\[ y_k=\lambda_k T(x_{n_k}) \]

is an unconditional basis in \(E\) for any permutation of the natural sequence \(\{n_k\}\), any sequence of numbers \(\lambda_k\ne 0\), and any isomorphism \(T:E\to E\). The bases \(\{x_k\}\) and \(\{y_k\}\) are called quasi-equivalent \((^{1,2})\).

A natural question arises: are all unconditional bases in \(E\) quasi-equivalent? This problem was solved positively for an entire class of nuclear countably normed spaces \((^{1-4})\). The assumption of nuclearity of the space was removed in \((^{5})\) for the case of so-called finite centers of Hilbert scales**. In the present paper the quasi-equivalence of all unconditional bases in infinite centers of completely continuous Hilbert scales is established. The proof of quasi-equivalence in infinite centers differs qualitatively from the arguments of \((^{5})\). This is due to the fact that Lemma 3 of \((^{5})\) has no analogue for infinite centers.

2°. Every positive self-adjoint bounded operator \(A\) in a Hilbert space \(H\) generates a Hilbert scale \(\{H_\alpha\}\) of Hilbert spaces \(H_\alpha\) with scalar products

\[ (x,y)_{H_\alpha}=(A^{-\alpha}x,A^{-\alpha}y), \]

where \((x,y)\) is the scalar product in \(H=H_0\) \((^{1,6})\). We shall call the scale \(\{H_\alpha\}\) completely continuous if the operator \(A\) is completely continuous. In the latter case one may assume that \(H_\alpha=l_2(a_k^\alpha)\), where \(a_k=1/\lambda_k\), \(\lambda_k\) are the eigenvalues of the operator \(A\), arranged in decreasing order, i.e. \(a_k\uparrow\infty\).

The countably Hilbert space

\[ E_\alpha(a_k)\equiv \lim_{\lambda<\alpha}\operatorname{pr} l_2(a_k^\lambda), \]

following \((^{1})\), will be called the center of the scale \(\{H_\lambda\}\)—finite if \(\alpha<\infty\), and infinite if \(\alpha=\infty\). For \(x=(\xi_1,\xi_2,\ldots,\xi_k,\ldots)\in H_\lambda\) we shall denote

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

If \(H_0\supset H_1\) are two Hilbert spaces with a (completely) continuous embedding, then there exists a (completely continuous) Hilbert scale \(\{G_\alpha\}\) such that \(G_0=H_0,\ G_1=H_1\). We shall say that \(\{G_\alpha\}\) is the scale generated by \(H_0\) and \(H_1\), and use the notation

\[ G_\alpha=(H_0)^{1-\alpha}(H_1)^\alpha. \]

We shall repeatedly use the following simple consequence of the known interpolation theorem \((^{7})\).

* This question was first posed by M. G. Haplanov for the space of analytic functions.

** Here and below the terminology of paper \((^{5})\) is used.

Lemma (cf. \((^9)\)). Let \(H_1 \supset H_2 \supset H_3 \supset H_4\) be a quadruple of Hilbert spaces with (complete) continuous embeddings. Then for any \(\alpha: 0 \leq \alpha \leq 1\) the following (complete) continuous embedding holds:
\[ (H_1)^{1-\alpha}(H_3)^\alpha \supset (H_2)^{1-\alpha}(H_4)^\alpha . \]

\(3^\circ\). We shall prove the main result.

Theorem. In the space \(E=E_\infty(a_k)\), \(a_k \uparrow \infty\), all unconditional bases are quasiequivalent.

Let \(\{x_k\}\) be an arbitrary unconditional basis in \(E\). It is required to show that \(\{x_k\}\) is quasiequivalent to the principal basis \(\{e_k\}\), consisting of the unit vectors. In other words, one has to find a permutation of the natural series \(\{n_k\}\) and a numerical sequence \(\sigma_k>0\) such that the bases \(\{y_k=\sigma_k x_{n_k}\}\) and \(\{e_k\}\) are equivalent, i.e. the linear operator \(T\) defined by the formula
\[ T\left(\sum \xi_k e_k\right)=\sum \xi_k y_k, \tag{1} \]
is an isomorphism of \(E\) onto itself.

Denote by \(\{x_k'\}\) the system biorthogonal to the basis \(\{x_k\}\). By Lemma 1 \((^5)\), the system of Hilbert norms
\[ |x|_{\lambda}^{(0)}=\left(\sum |x_k'(x)|^2 \,\|x_k\|_\lambda^2\right)^{1/2}, \qquad \lambda<\infty, \]
is equivalent to the original system of norms \(\{\|x\|_\lambda,\ \lambda<\infty\}\). Denote by \(H_\lambda^{(0)}\) the Hilbert space obtained from \(E\) by completion with respect to the norm \(\|x\|_\lambda^{(0)}\). From the equivalence of the systems of norms it follows that there exist numbers \(\mu_1<\nu_1<\nu_2<\mu_2,\ \lambda_1<\lambda_2\) such that the embeddings
\[ H_{\mu_1}\supset H_{\lambda_1}^{(0)}\supset H_{\nu_1}\supset H_{\nu_2}\supset H_{\lambda_2}^{(0)}\supset H_{\mu_2} \]
hold with continuous embeddings.

Take a permutation of the natural series \(\{n_k\}\) and a numerical sequence \(\sigma_k>0\) such that the system \(y_k=\sigma_k x_{n_k}\) satisfies the conditions
\[ \|y_k\|_{\lambda_1}^{(0)}=1,\qquad \|y_k\|_{\lambda_2}^{(0)}=b_k\uparrow\infty . \]

Below it will be shown that this permutation and renormalization are the desired ones, i.e. the operator (1) is an isomorphism of \(E\) onto itself.

Applying Lemma 2 \((^5)\) first to the quadruple of spaces
\[ H_{\mu_1}\supset H_{\lambda_1}^{(0)}\supset H_{\lambda_2}^{(0)}\supset H_{\mu_2} \]
and the bases \(\{a_k^{-\mu_1}e_k\}\), \(\{y_k\}\), and then to the quadruple of spaces
\[ H_{\lambda_1}^{(0)}\supset H_{\nu_1}\supset H_{\nu_2}\supset H_{\lambda_2}^{(0)} \]
and the bases \(\{y_k\}\), \(\{a_k^{-\nu_1}e_k\}\), we obtain the inequalities:
\[ a_k^{\nu_2-\nu_1}\prec b_k=\|y_k\|_{\lambda_2}^{(0)}\prec a_k^{\mu_2-\mu_1}. \tag{2} \]
Here and in what follows the notation \(a_k\prec b_k\) means \(a_k=O(b_k)\).

It follows from the lemma that for any \(\sigma: 0\leq \sigma\leq 1\) the continuous embeddings
\[ H_{\mu_1(1-\sigma)+\nu_2\sigma}\supset G_\sigma\supset H_{\nu_1(1-\sigma)+\mu_2\sigma},\qquad G_\sigma=(H_{\lambda_1}^{(0)})^{1-\sigma}(H_{\lambda_2}^{(0)})^\sigma \]
hold.

Hence, if one takes into account that \(\|y_k\|_{G_\sigma}=b_k^\sigma\), the inequalities \((\nu_1<\nu<\nu_2)\)
\[ b_k^{\sigma_2}\prec \|y_k\|_\nu\prec b_k^{\sigma_1}, \tag{3} \]
follow, where
\[ \sigma_1=\sigma_1(\nu)=(\nu-\nu_1)/(\mu_2-\nu_1), \qquad \sigma_2=\sigma_2(\nu)=(\nu-\mu_1)/(\nu_2-\mu_1). \]
Comparing inequalities (2) and (3), we obtain:
\[ a_k^{\gamma_2}\prec \|y_k\|_\nu\prec a_k^{\gamma_1}, \tag{4} \]
where
\[ \gamma_1=\sigma_1(\mu_2-\mu_1), \qquad \gamma_2=\sigma_2(\nu_2-\nu_1). \]

Now take an arbitrary \(\lambda>\lambda_2^{(0)}\) and \(\tau=\tau(\lambda)\ge \nu_2\), \(\chi=\chi(\lambda)\) such that the continuous embeddings hold:

\[ H_\tau \supset H_\lambda^{(0)} \supset H_\chi \]

and \(\tau(\lambda)\uparrow\infty\) as \(\lambda\uparrow\infty\).

Let \(\{y_{n_k}\}\) be a rearrangement of the basis \(\{y_k\}\) in decreasing order of the norms in \(H_\lambda^{(0)}\), i.e.,

\[ \|y_{n_k}\|_\lambda^{(0)}=s_k\uparrow\infty . \]

Application of Lemma 2 \((^5)\), analogously to the preceding one, gives the inequalities:

\[ a_k^{\tau-\nu_1}\prec \|y_{n_k}\|_\lambda^{(0)}=s_k\prec a_k^{\chi-\mu_1}. \tag{5} \]

As before, after applying the lemma of the present paper we obtain the estimates \((\nu_1<\nu<\tau)\):

\[ s_k^{\widetilde{\sigma}_2}\prec \|y_{n_k}\|_\nu \prec s_k^{\widetilde{\sigma}_1}, \tag{6} \]

where \(\widetilde{\sigma}_1=\widetilde{\sigma}_1(\nu,\lambda)=(\nu-\nu_1)/(\chi-\nu_1)\),
\(\widetilde{\sigma}_2=\widetilde{\sigma}_2(\nu,\lambda)=(\nu-\mu_1)/(\tau-\mu_1)\).

The following inequalities are the result of comparing inequalities (5) and (6):

\[ a_k^{\widetilde{\gamma}_2}\prec \|y_{n_k}\|_\nu \prec a_k^{\widetilde{\gamma}_1}, \tag{7} \]

where \(\widetilde{\gamma}_1=\widetilde{\sigma}_1(\chi-\mu_1)\), \(\widetilde{\gamma}_2=\widetilde{\sigma}_2(\tau-\nu_1)\).

Comparing inequalities (4) and (7), we obtain:

\[ a_{n_k}^{\gamma_2/\widetilde{\gamma}_1}\prec a_k\prec a_{n_k}^{\gamma_1/\widetilde{\gamma}_2}. \]

Together with inequality (5) this gives

\[ a_{n_k}^{\rho_2}\prec \|y_{n_k}\|_\lambda^{(0)}\prec a_{n_k}^{\rho_1}, \tag{8} \]

where

\[ \rho_1=\rho_1(\lambda)=\frac{\gamma_1}{\widetilde{\gamma}_2}(\chi-\mu_1) =\frac{(\nu-\nu_1)(\mu_2-\mu_1)(\tau-\mu_1)} {(\mu_2-\nu_1)(\nu-\mu_1)(\tau-\nu_1)}(\chi-\mu_1), \]

\[ \rho_2=\rho_2(\lambda)=\frac{\gamma_2}{\widetilde{\gamma}_1}(\tau-\nu_1) =\frac{(\nu-\mu_1)(\nu_2-\nu_1)(\chi-\nu_1)} {(\nu_2-\mu_1)(\nu-\nu_1)(\chi-\mu_1)}(\tau-\nu_1). \]

Obviously, as \(\lambda\uparrow\infty\),

\[ \rho_2(\lambda)\uparrow\infty . \tag{9} \]

Since \(\{n_k\}\) is a rearrangement of the natural sequence, formula (8) may be rewritten in the form

\[ C_2 a_k^{\rho_2}\le \|y_k\|_\lambda^{(0)}\le C_1 a_k^{\rho_1},\qquad C_1=C_1(\lambda)<\infty,\quad C_2=C_2(\lambda)>0 . \tag{10} \]

It is now easy to establish that the operator \(T\), defined by formula (1), is an isomorphism.

I take this opportunity to thank B. S. Mityagin, who drew my attention to the fact that a number of essential points in the proof of Theorem 1 in paper \((^5)\) had been used earlier (for another purpose) in \((^2)\) (see the proof of Theorem 11).

Rostov State University

Received
27 II 1969

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