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Reports of the Academy of Sciences of the USSR
- Volume 173, No. 1
UDC 513:88 + 513:83
MATHEMATICS
Yu. I. PETUNIN
PRENUCLEAR MAPPINGS IN SCALES OF BANACH AND HILBERT SPACES
(Presented by Academician A. N. Kolmogorov on 6 V 1966)
Let \(E\) and \(F\) be two Banach spaces with unit balls \(S_1(E)\) and \(S_1(F)\), respectively. Denote by \(S_1(E)^0\) the unit ball of the conjugate space \(E'\). A continuous linear mapping \(y = T(x)\), acting from the space \(E\) into \(F\), is called prenuclear (see \((^1)\)) if on \(S_1(E)^0\) there exists a positive Radon measure \(\mu\) such that
\[ \|Tx\| \leq \int_{S_1(E)^0} |\langle x,u\rangle|\,d\mu \quad \text{for all } x \in E \ (u \in E'). \tag{1} \]
The prenuclear norm \(\pi(T)\) for this mapping is defined by the formula
\[ \pi(T)=\inf \|\mu\|, \]
where the infimum is taken over all measures \(\mu\) satisfying condition (1).
By the symbol \(L^1(E)\) we shall everywhere in what follows denote the Banach space of absolutely convergent sequences \(\bar{x}=(x_1,x_2,\ldots,x_n,\ldots)\) of elements of the space \(E\) with norm
\[ \|\bar{x}\|_{L^1(E)}=\sum_{k=1}^{\infty}\|x_k\|_E, \]
and by \(l^1(E)\) we denote the space of all unconditionally convergent sequences \(\bar{x}=(x_1,x_2,\ldots,x_n,\ldots)\) \((x_n \in E)\), composed of elements of \(E\), where
\[ \|\bar{x}\|_{l^1(E)} = \sup_{u\in S_1(E)^0} \sum_{k=1}^{\infty} |\langle x_k,u\rangle|. \]
It is said that the mapping \(y=T(x)\) \((y\in F,\ x\in E)\) is absolutely summing if \(T\) maps every unconditionally convergent sequence \(x_n \in E\) into an absolutely convergent one.
In \((^1)\) it is shown that the class of all prenuclear mappings \(y=T(x)\) \((y\in F,\ x\in E)\) coincides with the set of absolutely summing mappings; moreover
\[ \pi(T)=\sup \sum_{k=1}^{\infty}\|Tx_k\|_F, \]
where the supremum is taken over all elements \(x_k \in E\) for which
\[ \sup_{u\in S_1(E)^0} \sum_{k=1}^{\infty} |\langle x_k,u\rangle| \leq 1. \]
It follows from this that every prenuclear mapping \(T:E\to F\) generates a mapping \(\overline{T}: l^1(E)\to L^1(F)\) by the formula
\[ \bar{y}=\overline{T}(\bar{x})=\{Tx_k\}_{k=1}^{\infty}, \]
and the prenuclear norm \(\pi(T)\) is the usual norm of the mapping \(\overline{T}\).
Let \(F_\alpha\) \((0 \leq \alpha \leq 1)\) be an arbitrary normal scale of Banach spaces (see (2)). It is easy to see that in this case the family of spaces \(L^1(F_\alpha)\) also forms a normal scale.
Proposition 1. If \(F_\alpha^{\min}\) \((0 \leq \alpha \leq 1)\) is the minimal scale (see (2)) joining the spaces \(F_0\) and \(F_1\), then the scale \(L^1(F_\alpha^{\min})\) majorizes (see (2)) the minimal scale constructed from the spaces \(L^1(F_0)\) and \(L^1(F_1)\).
Proposition 2. The family of spaces \(L^1(F_\alpha^{\max})\) forms the maximal scale (see (2)) joining the space \(L^1(F_0)\) with \(L^1(F_1)\), if the scale \(F_\alpha^{\max}\) is maximal.
Remark 1. It follows from Propositions 1 and 2 that the scale \(L^1(F_\alpha^{\min})\) has the normal interpolation property for linear operators (see (2)) with respect to any minimal scale, while the scale \(L^1(F_\alpha^{\min})\) is strictly interpolation with respect to any normal scale. We note that the spaces \(L^1(F_\alpha^{\min})\) do not necessarily form a minimal scale: it may happen that the scale \(L^1(F_\alpha^{\min})\) is maximal.
Proposition 3. For any normal scale \(E_\alpha\) \((0 \leq \alpha \leq 1)\), the spaces \(l^1(E_\alpha)\) form a normal scale.
Proof. Indeed,
\[ \varphi(\alpha)=\|\bar{x}\|_{l^1(E_\alpha)} = \sup_{u \in S_1(E_\alpha)^0}\sum_{k=1}^{\infty} |\langle x_k,u\rangle| = \]
\[ = \sup_{\substack{|\theta_k|=1\\ u\in S_1(E_\alpha)^0}} \sum_{k=1}^{\infty}\langle \theta_k x_k,u\rangle = \sup_{\substack{|\theta_k|=1\\ u\in S_1(E_\alpha)^0}} \left\langle \sum_{k=1}^{\infty}\theta_k x_k,u\right\rangle = \sup_{|\theta_k|=1}\left\|\sum_{k=1}^{\infty}\theta_k x_k\right\|_{E_\alpha}. \]
Hence the logarithmic convexity of the function \(\varphi(\alpha)\) follows, since the supremum of logarithmically convex functions is a logarithmically convex function.
The remaining axioms of a normal scale are obvious.
Theorem 1. A normal scale of Banach spaces \(l^1(E_\alpha)\) \((0 \leq \alpha \leq 1)\) is regular if and only if the family of spaces \(E_\alpha\) \((0 \leq \alpha \leq 1)\) forms a regular scale (see (2)).
The proof of Theorem 1 is based on properties of Pietsch products.
Definition 1. Let \(E_1,E_2,\ldots,E_n\) be subspaces of a Banach space \(E\). The Pietsch product \(\pi(E_1,\ldots,E_n)\) of the spaces \(E_1,\ldots,E_n\) is the direct product \(E_1\times\cdots\times E_n\), endowed with the norm
\[ \|\tilde{x}\|_{\pi(E_1\ldots E_n)} = \|(x_1,\ldots,x_n)\|_{\pi(E_1\ldots E_n)} = \sup_{\theta^1_k} \left\|\sum_{k=1}^{\infty}\theta_k x_k\right\|_E, \]
where \(\tilde{x}\in E_1\times\cdots\times E_n\) and \(x_k\in E_k\) \((k=1,\ldots,n)\).
In particular, if \(E_1=E_2=\cdots=E_n=E\), then the space \(\pi(E_1\ldots E_n)=\pi(E^n)\) is called the \(n\)-th Pietsch power of the Banach space \(E\).
It is not difficult to note that for any normal scale \(E_\alpha\) \((0 \leq \alpha \leq 1)\), the family of spaces \(\pi(E_\alpha^n)\) forms a normal scale, which we shall call the \(n\)-th Pietsch power of the scale \(E_\alpha\).
Lemma 1. The Pietsch square \(\pi(E_\alpha^2)\) of a regular scale \(E_\alpha\) \((0 \leq \alpha \leq 1)\) is a regular scale.
Proof. The unit ball \(S_1[\pi(E_\alpha^2)]^0\) of the conjugate space \(\pi(E_\alpha^2)'\) is closed in the topology \(\sigma(\pi(E_\alpha^2)',\pi(E_\alpha^2))\)
by the convex hull of the sets
\[ \mathcal U=\{(u,u):\ \|u\|_{E_\alpha'}=1\},\qquad \mathcal V=\{(v,-v):\ \|v\|_{E_\alpha'}=1\}, \]
since the ball \(S_1[\pi(E_\alpha^2)]^0\) coincides with the polar of the set
\[ \begin{aligned} S_1[\pi(E_\alpha^2)] &=\left\{\widetilde x=(x_1,x_2):\ \sup_{|\theta_k|=1} \|\theta_1x_1+\theta_2x_2\|_{E_\alpha}\leqslant 1;\ x_1,x_2\in E_\alpha\right\}\\ &=\left\{\widetilde x:\ \sup_{u\in S_1(E_\alpha)^0} \left[|\langle x_1,u\rangle|+|\langle x_2,u\rangle|\right]\leqslant 1\right\}. \end{aligned} \]
Let us show that all boundary points \(\widetilde w_2\) of the set \(S_1[\pi(E_\alpha^2)]^0\) can be represented in the form
\[ \widetilde w_2=\mu\widetilde u_2+\nu\widetilde v_2 \qquad(\mu+\nu=1;\ \mu,\nu\geqslant0), \tag{2} \]
where \(\widetilde u_2\) and \(\widetilde v_2\) run over the boundaries of the sets \(\mathcal U\) and \(\mathcal V\), respectively. To this end consider the subspaces
\[ \mathcal L=\{(u,u):\ u\in E_\alpha'\}\subset \pi(E_\alpha^2)',\qquad \mathcal M=\{(v,-v):\ v\in E_\alpha'\}\subset \pi(E_\alpha^2)', \]
each of which is isometrically conjugate to the space \(E_\alpha'\), for the sets \(\mathcal U\) and \(\mathcal V\) are the unit balls in the subspaces \(\mathcal L\) and \(\mathcal M\). The subspaces \(\mathcal L\) and \(\mathcal M\), being the polars of the subspaces \(\{\widetilde x:\ (x,-x),\ x\in E_\alpha\}\) and \(\{\widetilde y:\ (y,y),\ y\in E_\alpha\}\), are closed in the topology \(\sigma(E_\alpha'\times E_\alpha',E_\alpha\times E_\alpha)\). The unit ball of the conjugate space is a compact set in the weak topology; therefore the convex hull \(\mu\mathcal U\oplus\mathcal V\) \((\mu+\nu=1,\ \mu,\nu\geqslant0)\) is compact in the topology \(\sigma(E_\alpha'\times E_\alpha',E_\alpha\times E_\alpha)\), so that \(S_1[\pi(E_\alpha^2)]^0=\mu\mathcal U\oplus\nu\mathcal V\) (see (3), Chapter II, §4, Proposition 1).
Consequently, to prove the representation (2) it is enough for us to show that the point \(\mu\widetilde u_2+\nu\widetilde v_2\) \((\mu+\nu=1;\ \mu,\nu\geqslant0)\) will be a boundary point of the ball \(S_1[\pi(E_\alpha')]^0\).
Choose an arbitrary number \(\theta>1\), as close to one as desired. The intersection of the subspaces \(\mathcal L\) and \(\mathcal M\) contains only one zero element; in view of this circumstance there exist two elements \(\widetilde x_0,\widetilde y_0\in E_\alpha\times E_\alpha\) for which the conditions
\[ \begin{gathered} \langle \widetilde x_0,\theta\widetilde u_2\rangle>1,\qquad \langle \widetilde x_0,\widetilde v\rangle=0\quad\text{for all }\widetilde v\in\mathcal M,\\ \langle \widetilde y_0,\theta\widetilde v_2\rangle<1\qquad (\widetilde y_0,\widetilde u_0)=0\quad\text{for all }\widetilde u\in\mathcal L,\\ \|\widetilde x_0\|_{\pi(E_\alpha^2)}=\|\widetilde y_0\|_{\pi(E_\alpha^2)}=1. \end{gathered} \]
Consider the hyperplane \(\langle \widetilde x_0+\widetilde y_0,\widetilde u\rangle=1,\ \widetilde u\in\pi(E_\alpha^2)'\), which is, obviously, closed in the weak topology; therefore the weakly closed convex hull of the sets \(\mathcal U\) and \(\mathcal V\) lies on one side of this hyperplane. Denote by \(l\) the straight line belonging to the two-dimensional plane \(P\) generated by the elements \(\widetilde u_2,\widetilde v_2\) and passing through the points \(\theta\widetilde u_2,\theta\widetilde v_2\). Let \(\widetilde w\) be an arbitrary point of the segment joining the points \(\widetilde u_2\) and \(\widetilde v_2\), and let \(W\) be a neighborhood of the point \(\widetilde w\) in the plane \(P\). This neighborhood, for some \(\theta>1\), contains points of the straight line \(l_0\); hence \(\widetilde w\) is a boundary point of \(S_1[\pi(E_\alpha^2)]^0\).
Now consider an arbitrary element \(\widetilde u\in\pi(E_0^2)'\). Let \(\widetilde u_2\) and \(\widetilde v_2\) be two elements of the sets \(\pi(E_\alpha^2)'\cap\mathcal U\), \(\pi(E_0^2)'\cap\mathcal V\), such that \(\widetilde u=\mu\widetilde u_2+\nu\widetilde v_2\) \((\mu+\nu=1;\ \mu,\nu=0)\).
On the basis of the preceding arguments we may assert the validity of the equality
\[ \|\widetilde u\|_{\pi(E_\alpha^2)'} =\mu\|\widetilde u_2\|_{\pi(E_\alpha^2)'} +\nu\|\widetilde v_2\|_{\pi(E_\alpha^2)'}, \]
for every \(\alpha\in[0,1]\). The function \(f(\alpha)=\|\widetilde u\|_{\pi(E_\alpha^2)'}\) will be logarithmically convex as the sum of two logarithmically convex functions.
Lemma 2. The \(n\)-th power of Pisier \(\pi(E_\alpha^n)\) of a regular scale \(E_\alpha\) \((0 \leq \alpha \leq 1)\) is a regular scale.
Using the almost interpolation property of regular scales (see (2)) and Theorem 1, we obtain that the following is valid.
Theorem 2. Let \(y = T(x)\) be a prenuclear operator mapping the spaces \(E_0\) into \(F_0\) and \(E_1\) into \(F_1\); let \(E_\alpha\) \((0 \leq \alpha \leq 1)\) be a regular scale connecting the space \(E_0\) with \(E_1\). Then, for an arbitrary normal scale \(F_\alpha\) \((0 \leq \alpha \leq 1)\), the operator \(y = T(x)\) is a prenuclear mapping acting from the space \(E_\beta\) into \(F_\alpha\) for \(\beta > \alpha\).
A prenuclear mapping acting in a Hilbert space is a Hilbert—Schmidt mapping (see (1)).
Theorem 3. Let \(H_\alpha\) \((0 \leq \alpha \leq 1)\) be a Hilbert scale (see (2)) and \(G_\alpha\) \((0 \leq \alpha \leq 1)\) an arbitrary scale of Hilbert spaces. If \(y = A(x)\) is a Hilbert—Schmidt operator acting from the spaces \(H_0\) into \(G_0\), \(H_1\) into \(G_1\), then \(A\) will be a Hilbert—Schmidt operator mapping the space \(H_\alpha\) into \(G_\alpha\), and the Hilbert—Schmidt norm \(|A|_{H_\alpha \to G_\alpha}\) is logarithmically convex as a function of \(\alpha\).
In conclusion, the author takes the opportunity to express gratitude to S. G. Krein and B. S. Mityagin for their attention and valuable comments.
Voronezh State
University
Received
6 V 1966
REFERENCES
- A. Pietsch, Sborn. per. Matematika, 8, 2, 77 (1964).
- S. G. Krein, Yu. I. Petunin, UMN, 21, 2 (127) (1966).
- N. Bourbaki, Topological Vector Spaces, Moscow, 1959.