UDC 513.88:513.83

MATHEMATICS

Yu. I. PETUNIN

THE RELATEDNESS OF THREE BANACH SPACES

(Presented by Academician A. N. Kolmogorov on 17 XII 1965)

In this paper the symbol \(E_1 \subset E_0\) is used to denote the Banach space \(E_1\), normally embedded in the Banach space \(E_0\) (see \((^1)\)). Denote by \(W_\alpha\) \((0 \leq \alpha \leq 1)\) the convex hull of the set

\[ S_\alpha=\{x:\|x\|_{E_0}^{1-\alpha}\|x\|_{E_1}^{\alpha}\leq 1,\quad x\in E_1\}. \]

Definition 1. Let \(E_0 \supset E \supset E_1\). The index of the intermediate space \(E\) with respect to the spaces \(E_0\) and \(E_1\) is the number

\[ \alpha_i=\inf_{W_\alpha\subset S_1(E)} \alpha=\operatorname{ind}_{E_0,E_1}(E), \]

where \(S_1(E)\) is the unit ball of the space \(E\).

Definition 2. We shall say that a continuous normal scale (see \((^2)\)) \(E_\alpha\) \((0 \leq \alpha \leq 1)\) connects the spaces \(E_0 \supset E \supset E_1\), if

\[ E_{\alpha=0}=E_0,\qquad E_{\alpha=\operatorname{ind}_{E_0,E_1}(E)}=E,\qquad E_{\alpha=1}=E_1. \]

Definition 3. Three normally embedded Banach spaces \(E_0 \supset E \supset E_1\) are called related if there exists a continuous normal scale \(E_\alpha\) \((0 \leq \alpha \leq 1)\) connecting these spaces.

Consider the class \(K(E_0,E_1)\) of continuous normal scales \(E_\alpha\) \((0 \leq \alpha \leq 1)\) which connect the spaces \(E_0 \supset E_1\) and majorize the minimal scale constructed from the spaces \(E_0\) and \(E_1\) (see \((^3)\)).

Definition 4. By absolutely related Banach spaces \(E_0,E,E_1\) we shall mean normally embedded spaces \(E_0 \supset E \supset E_1\) for which any two normal scales \(E_\alpha \in K(E_0,E)\) \((0 \leq \alpha \leq \operatorname{ind}_{E_0,E_1}(E))\) and \(\hat E_\alpha \in K(E,E_1)\) \((\operatorname{ind}_{E_0,E_1}(E) \leq \alpha \leq 1)\), when glued at the point \(\alpha=\operatorname{ind}_{E_0,E_1}(E)\), form a continuous normal scale (see \((^{1,2})\)).

Lemma 1. In order that three normally embedded Banach spaces \(E_0 \supset E \supset E_1\) be related, it is necessary and sufficient that, upon gluing the maximal scales \(E_\alpha^{\max}\) \((0 \leq \alpha \leq \alpha_i,\ E_{\alpha=\alpha_i}=E)\) and \(\hat E_\alpha^{\max}\) \((\alpha_i \leq \alpha \leq 1,\ \hat E_{\alpha_i}=E,\ \hat E_{\alpha=1}=E_1)\) at the point \(\alpha=\operatorname{ind}_{E_0,E_1}(E)\), a continuous normal scale be obtained (see \((^{1,2})\)).

Lemma 2. Three normally embedded Banach spaces \(E_0 \supset E \supset E_1\) are absolutely related if and only if the family of Banach spaces

\[ E_\alpha= \begin{cases} E_\alpha^{\min}(E_0,E); & 0 \leq \alpha \leq \alpha_i,\ E_{\alpha=\alpha_i}=E,\\ E_\alpha^{\min}(E,E_1); & \alpha_i \leq \alpha \leq 1, \end{cases} \]

where \(E_\alpha^{\min}(E_0,E)\), \(E_\alpha^{\min}(E,E_1)\) are the minimal scales constructed from the spaces \(E_0 \supset E\) and \(E \supset E_1\), respectively, is a continuous normal scale (see \((^3)\)).

We shall formulate a criterion for the absolute relatedness of three normally embedded Banach spaces \(E_0 \supset E \supset E_1\), restricting ourselves, for simplicity, to the case of a smooth unit sphere of the space \(E\) (see (4), p. 187, Definition 1).

Theorem 1. In order that three normally embedded Banach spaces \(E_0 \supset E \supset E_1\) be absolutely related, it is necessary and sufficient that, for all functionals \(f(x) \in E_0'\), the inequality
\[ \|f\|_{E'} \geq \|f\|_{E_0'}^{\,1-\alpha_i}\|f\|_{E_1'}^{\,\alpha_i}. \tag{1} \]

Proof of sufficiency. On the basis of the definition of the norm of an element \(x\) in the minimal scale,
\[ \|x\|_{E^{\min}_{\alpha}(E_0,E)} = \sup_{f\in E_0'} \frac{|f(x)|}{\|f\|_{E_0'}^{(\alpha_i-\alpha)/\alpha_i}\|f\|_{E'}^{\alpha/\alpha_i}} \qquad (x\in E_1,\ 0\leq \alpha\leq \alpha_i), \]
\[ \|x\|_{E^{\min}_{\alpha}(E,E_1)} = \sup_{f\in E'} \frac{|f(x)|}{\|f\|_{E'}^{(1-\alpha)/(1-\alpha_i)}\|f\|_{E_1'}^{(\alpha-\alpha_i)/(1-\alpha_i)}} \qquad (x\in E_1,\ \alpha_i\leq \alpha\leq 1). \]

Consider the function
\[ \varphi(\alpha,x)= \begin{cases} \ln\|x\|_{E^{\min}_{\alpha}(E_0,E)}, & \text{if } 0\leq \alpha\leq \alpha_i,\\[4pt] \ln\|x\|_{E^{\min}_{\alpha}(E,E_1)}, & \text{if } \alpha_i\leq \alpha\leq 1. \end{cases} \]

It is not hard to see that the absolute relatedness of the Banach spaces \(E_0 \supset E \supset E_1\) is equivalent to the inequality
\[ \varphi_\alpha'(\alpha_i-0,x)\leq \varphi_\alpha'(\alpha_i+0,x),\qquad x\in E_1, \]
where \(\varphi_\alpha'(\alpha_i-0,x)\), \(\varphi_\alpha'(\alpha_i+0,x)\) are the left and right derivatives with respect to \(\alpha\) of the function \(\varphi(\alpha,x)\) at the point \(\alpha=\alpha_i\).

Suppose that inequality (1) is satisfied for all elements \(f\in E_0'\). By virtue of the smoothness of the space \(E_0 \supset E\), we can choose a sequence of functionals \(\{f_n\}\in E_0'\) such that
\[ \lim_{n\to\infty}|f_n(x)|/\|f_n\|_{E'}=\|x\|_E \tag{2} \]
(see (2), Theorem 1).

Denote by
\[ \Phi_n(\alpha,x)= \ln \frac{|f_n(x)|} {\|f_n\|_{E_0'}^{(\alpha_i-\alpha)/\alpha_i}\|f\|_{E'}^{\alpha/\alpha_i}} \qquad (0\leq \alpha\leq \alpha_i,\ x\in E_1), \]
\[ \Psi_n(\alpha,x)= \ln \frac{|f_n(x)|} {\|f_n\|_{E'}^{(1-\alpha)/(1-\alpha_i)}\|f_n\|_{E_1'}^{(\alpha-\alpha_i)/(1-\alpha_i)}} \qquad (\alpha_i\leq \alpha\leq 1,\ x\in E_1). \]

It is easy to see that
\[ \varphi(\alpha,x)\geq \Phi_n(\alpha,x)\qquad (0\leq \alpha\leq \alpha_i), \]
\[ \varphi(\alpha,x)\geq \Psi_n(\alpha,x)\qquad (\alpha_i\leq \alpha\leq 1), \tag{3} \]
moreover, without loss of generality one may assume the existence of the limits
\[ \lim_{n\to\infty}[\Phi_n(\alpha_i,x)]_\alpha',\qquad \lim_{n\to\infty}[\Psi_n(\alpha_i,x)]_\alpha'. \]

From conditions (2) and (3) follow the inequalities
\[ \varphi_\alpha'(\alpha_i-0,x)\leq \lim_{n\to\infty}[\Phi_n(\alpha_i,x)]_\alpha', \]
\[ \varphi_\alpha'(\alpha_i+0,x)\geq \lim_{n\to\infty}[\Psi_n(\alpha_i,x)]_\alpha'. \]

Finally, by virtue of relation (1),
\[ \frac{\|f_n\|_{E_0'}^{1/\alpha_i}}{\|f_n\|_{E_1'}^{1/\alpha_i}} = [\Phi_n(\alpha_i,x)]_\alpha' \leq \frac{\|f_n\|_{E'}^{1/(1-\alpha_i)}}{\|f_n\|_{E_1'}^{1/(1-\alpha_i)}} = [\Psi_n(\alpha_i,x)]_\alpha', \]

so that

\[ \varphi_{\alpha}'(\alpha_i-0,x)\leq \varphi_{\alpha}'(\alpha_i+0,x). \]

The sufficiency of the conditions of Theorem 1 is proved.

We proceed to present the scheme of the proof of necessity. First one proves

Lemma 3. Let \(E_0 \supset E_1\) be normally embedded Banach spaces, \(L\) a linear manifold of \(E_1\), and \(L_{E_0}, L_{E_1}\) the linear manifold \(L\) endowed with the norms \(\|x\|_{E_0}\) and \(\|x\|_{E_1}\), respectively.

Then for every element \(x \in L\)

\[ \|x\|_{E_\alpha^{\min}(L_{E_0},L_{E_1})} \geq \|x\|_{E_\alpha^{\min}(E_0,E_1)} \qquad (0\leq \alpha \leq 1), \]

where \(E_\alpha^{\min}(L_{E_0},L_{E_1})\), \(E_\alpha^{\min}(E_0,E_1)\) \((0\leq \alpha \leq 1)\) are the minimal scales constructed from the spaces \(L_{E_0}\supset L_{E_1}\) and \(E_0\supset E_1\).

Moreover, one can show that from the absolute relatedness of the finite-dimensional spaces \(E_0\supset E\supset E_1\) there follows inequality (1), under the condition that the unit sphere of the space \(E\) is smooth.

Let \(f(x)\) be an arbitrary functional from the space \(E_0'\). By assumption, the space \(E_1\) is normally embedded in \(E\) and \(E_0\); therefore, for any \(\delta>0\) there exist elements \(x_0,x,x_1\in E_1\) with norms \(\|x_0\|_{E_0}=\|x\|_E=\|x_1\|_{E_1}=1\), for which

\[ |f(x_0)|=\|f\|_{E_0'}+\varepsilon_0,\qquad |f(x)|=\|f\|_{E'}+\varepsilon,\qquad |f(x_1)|=\|f\|_{E_1'}+\varepsilon_1, \tag{4} \]

where \(\varepsilon_0,\varepsilon,\varepsilon_1<\delta\).

Denote by \(L_{E_0}, L_E, L_{E_1}\) the subspaces of \(E_0,E\), and \(E_1\) generated by the elements \(x_0,x,x_1\). Then

\[ L_{E_0}\supset L_E\supset L_{E_1}, \]

and the unit sphere of the space \(L_E\) is smooth. On the basis of Lemmas 2 and 3 we may assert that the Banach spaces \(L_{E_0}, L_E, L_{E_1}\) are absolutely related; consequently, the norm \(\|\bar f\|_{L_E'}\) of the functional \(\bar f(x)\), induced on \(L_{E_0}\) by the functional \(f(x)\), satisfies the inequality

\[ \|\bar f\|_{L_E'} \geq \|\bar f\|_{L_{E_0}'}^{\,1-\alpha_i} \|\bar f\|_{L_{E_1}'}^{\,\alpha_i}. \]

Taking into account the equalities (4), we obtain

\[ \|\bar f\|_{L_{E_0}'}=\|f\|_{E_0'}+\hat\varepsilon_0,\qquad \|\bar f\|_{L_{E_1}'}=\|f\|_{E_1'}+\hat\varepsilon_1, \]

\[ \|\bar f\|_{L_E'}=\|f\|_{E'}+\hat\varepsilon,\qquad \text{where }\hat\varepsilon_0,\hat\varepsilon,\hat\varepsilon_1<\delta. \]

Thus, the relation

\[ \|f\|_{E'}+\hat\varepsilon \geq (\|f\|_{E_0'}+\hat\varepsilon_0)^{1-\alpha_i} (\|f\|_{E_1'}+\hat\varepsilon_1)^{\alpha_i} \]

holds for all \(\hat\varepsilon_0,\hat\varepsilon,\hat\varepsilon_1<\delta\), and therefore

\[ \|f\|_{E'}\geq \|f\|_{E_0'}^{\,1-\alpha_i} \|f\|_{E_1'}^{\,\alpha_i}. \]

The theorem is proved.

Let \(E_0\supset E\supset E_1\) be three normally embedded Banach spaces possessing the properties: 1) the spaces \(E_0\supset E\) and \(E\supset E_1\) are related; 2) \(\operatorname{ind}_{E_0,E_1}(E)\ne 1\).

It is natural to consider the following problem: will the spaces \(E_0\supset E\supset E_1\) be related if conditions 1) and 2) are satisfied for them? We shall show that, in this general formulation, the indicated problem is answered in the negative. Indeed, it is not difficult to construct two finite-dimensional normally embedded Banach spaces \(E_0\supset E_1\), for which the family of Ba

Banach spaces \(\widetilde E_\alpha(E_0,E_1)\) with unit balls \(W_\alpha\) \((0\leq \alpha \leq 1)\) does not form a normal scale.

Since the maximal scale \(E_\alpha^{\min}(E_0,E_1)\) joining the spaces \(E_0\) and \(E_1\) does not coincide with the scale \(\widetilde E_\alpha(E_0,E_1)\), there exists a number \(\alpha_i \ne 1\) such that

\[ \widetilde E_{\alpha_i}(E_0,E_1)\subset E_{\alpha_i}^{\max}(E_0,E_1), \tag{5} \]

but

\[ \widetilde E_{\alpha_i}(E_0,E_1)\ne E_{\alpha_i}^{\max}(E_0,E_1). \tag{6} \]

The spaces \(E_0 \supset E_{\alpha_i}(E_0,E_1)\supset E_1\) give an example of three non-kindred normally embedded Banach spaces. Indeed, if one assumes that the spaces \(E_0,\widetilde E_{\alpha_i}(E_0,E_1),E_1\) are kindred, then

\[ \widetilde E_{\alpha_i}(E_0,E_1)\supset E_{\alpha_i}^{\max}(E_0,E_1). \]

This latter embedding, together with (5), contradicts condition (6).

Denote by \(\mathscr L_1(\rho)\) the Banach space of measurable functions defined on the interval \([0,1]\) and summable with almost everywhere positive weight \(\rho(t)\in L_1\).

The space \(E_1=\mathscr L_1(\rho_1)\) is normally embedded in \(E_0=\mathscr L_1(\rho_0)\) if and only if \(\rho_0(t)\leq \rho_1(t)\) for almost all \(t\in[0,1]\).

Theorem 2. The maximal scale joining the spaces \(\mathscr L_1(\rho_0)=E_0\) and \(\mathscr L_1(\rho_1)=E_1\) consists of the spaces \(\mathscr L_1(\rho_\alpha)\) with weights

\[ \rho_\alpha(t)=[\rho_0(t)]^{1-\alpha}[\rho_1(t)]^\alpha . \]

The index of the intermediate space \(\mathscr L_1(\rho)\) between the spaces \(\mathscr L_1(\rho_0)\) and \(\mathscr L_1(\rho_1)\) is equal to

\[ \operatorname{ind}_{\mathscr L_1(\rho_0),\,\mathscr L_1(\rho_1)}[\mathscr L_1(\rho)] = \inf_{\rho(t)\leq[\rho_0(t)]^{1-\alpha}[\rho_1(t)]^\alpha}\alpha . \]

Theorem 3. Let \(\mathscr L_1(\rho_0)\supset \mathscr L_1(\rho)\supset \mathscr L_1(\rho_1)\) be Banach spaces of measurable functions defined on the interval \([0,1]\) and integrable with almost everywhere positive summable weights \(\rho_0(t),\rho(t),\rho_1(t)\).

In order that the Banach spaces \(\mathscr L_1(\rho_0),\mathscr L_1(\rho),\mathscr L_1(\rho_1)\) be kindred, it is necessary and sufficient that

\[ \operatorname{ind}_{\mathscr L_1(\rho_0),\mathscr L_1(\rho_1)}[\mathscr L_1(\rho)]=\alpha_i\ne 1. \]

The author takes this opportunity to express gratitude to S. G. Krein for useful advice and valuable discussions.

Voronezh State University

Received
17 XII 1965

References

1. S. G. Krein, DAN, 132, 510 (1960).
2. S. G. Krein, Yu. I. Petunin, DAN, 139, 1295 (1961).
3. S. G. Krein, Yu. I. Petunin, DAN, 154, 30 (1964).
4. M. M. Day, Linear Normed Spaces, IL, 1961.