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Reports of the Academy of Sciences of the USSR
1962, Volume 145, No. 6
MATHEMATICS
Ya. B. Rutitskii
ON LINEAR OPERATORS IN ORLICZ COORDINATE SPACES
(Presented by Academician P. S. Aleksandrov on 28 III 1962)
Let \(P(u)\) and \(Q(u)\) \((-\infty<u<\infty)\) be continuous, even functions, monotonically increasing for \(u\geq 0\), with \(P(u)=Q(u)=0\). Let \(\{a_{ik}\}_{1}^{\infty}\) be an infinite matrix satisfying the conditions
\[ \sum_{k=1}^{\infty} P(a_{ik}) \leq C_1,\qquad \sum_{i=1}^{\infty} Q(a_{ik}) \leq C_2 . \tag{1} \]
We study the conditions for continuity and complete continuity of the linear matrix operator
\[ y=Ax, \tag{2} \]
where \(x=(\xi_1,\ldots,\xi_k,\ldots)\), \(y=(\eta_1,\ldots,\eta_i,\ldots)\), and
\[ \eta_i=\sum_{k=1}^{\infty} a_{ik}\xi_k, \tag{3} \]
as an operator acting from the Orlicz coordinate space \(l_M\) \((^1)\) into some Orlicz coordinate space \(l_{M_2}\).
A number of conditions for continuity and complete continuity of the operator under consideration were obtained by Yu. I. Gribanov \((^2,^3)\) and others. Here new conditions are formulated.
§ 1. We first give several auxiliary assertions.
Lemma. In order that the \(N\)-function \(M_1(u)\) satisfy the \(\Delta_2\)-condition for small \(u\) (for all \(u\)), it is necessary and sufficient that there exist an \(N\)-function \(M_2(v)\) such that
\[ M_1(auv)\geq M_1(u)M_2(v) \tag{4} \]
for small \(u\) (for all \(u\)) and small \(v\).
The element \(z=(\xi_1\eta_1,\ldots,\xi_k\eta_k,\ldots)\) will be called the \(\odot\)-product of the elements \(x=(\xi_1,\ldots,\xi_k,\ldots)\) and \(y=(\eta_1,\ldots,\eta_k,\ldots)\), and will be denoted by \(x\odot y\).
Theorem 1. In order that the \(\odot\)-product \(x\odot y\) belong to the space \(l_{M_1}\) for every pair of elements \(x\in l_{M_2}\) and \(y\in l_{M_3}\), it is necessary and sufficient that the \(N\)-functions \(M_1(u)\), \(M_2(u)\), and \(M_3(u)\) satisfy the relation
\[ M_1(auv)\leq M_2(u)+M_3(v) \tag{5} \]
for some \(a>0\) and all \(u,v\leq u_0\).
An analogous theorem for Orlicz function spaces was proved by T. Ando \((^4)\). The assertions of these theorems are valid for any finite number of factors.
Let \(\mathfrak{M}\) be the class of coordinate Banach spaces \(E\) possessing the following properties:
-
From the condition \(\lim_{n\to\infty}\|x_n\|=0\) it follows that \(x_n\) converges to zero coordinatewise.
-
From the condition \(\lim_{n\to\infty}\|x_n\|=0\) there follows the existence in \(E\) of such an ele-
element \(x_0\) and such a subsequence \(x_{n_p}\) \((p=1,2,\ldots)\) that \(|\xi_k^{(n_p)}| \leq |\xi_k^0|\) for all \(p\) and \(k\) (\(\xi_k^{(n)}\) is the \(k\)-th coordinate of the element \(x_n\)).
- From the fact that the sequence \(x_n \in E\) converges coordinatewise to \(x\), it follows that
\[ \|x\| \leq \lim_{n\to\infty} \|x_n\|. \]
Theorem 2. Let \(E\) and \(E_1\) be spaces of the class \(\mathfrak M\). Let the matrix \(\{a_{ik}\}_1^\infty\) define the operator (2)—(3) acting from \(E\) into \(E_1\). Then the operator (2)—(3) is continuous.
This theorem is an analogue of a theorem of Banach ([5], pp. 74–75) for spaces of measurable functions.
It is easy to see that Orlicz coordinate spaces belong to the class \(\mathfrak M\).
§ 2. We shall say that the function \(\Phi_1(u)\) is less than the function \(\Phi_2(u)\) (and write \(\Phi_1(u) < \Phi_2(u)\)) if \(\Phi_1(u) \leq \Phi_2(\omega u)\) for \(u \leq u_0\). We shall say that \(\Phi_1(u)\) is essentially less than \(\Phi_2(u)\) (and write \(\Phi_1(u) \ll \Phi_2(u)\)) if
\[
\lim_{u\to\infty}\frac{\Phi_1(u)}{\Phi_2(\varepsilon u)}=0
\]
for every \(\varepsilon>0\). The functions \(\Phi_1(u)\) and \(\Phi_2(u)\) are called equivalent \((\Phi_1(u)\sim \Phi_2(u))\) if simultaneously \(\Phi_1(u)<\Phi_2(u)\) and \(\Phi_2(u)<\Phi_1(u)\).
A simple consequence of Hölder’s inequality for Orlicz coordinate spaces is the following.
Theorem 3. If the second of conditions (1) is satisfied, then the operator (2)—(3) acts continuously from the space \(l_1\) of summable sequences into any space \(l_{M_1}\) for which \(M_1(u)<Q(u)\).
Theorem 4. If the series
\[
\sum_{i=1}^{\infty} Q(\lambda a_{ik})
\]
converge uniformly with respect to \(k\) for every \(\lambda>1\), then under the hypotheses of Theorem 3 the operator \(A\) acts from \(l_1\) into \(h_{M_1}\) and is completely continuous.
Here \(h_{M_1}\) denotes the closure in \(l_{M_1}\) of the set of elements with a finite number of coordinates different from zero.
Everywhere below it is assumed that the monotone functions under consideration possess the following property: from the condition \(\Phi_1(u)\ll\Phi_2(u)\) it follows that the functions \(\Phi_1(u)/\Phi_2(u)\) and \(\Phi_2^{-1}(v)/\Phi_1^{-1}(v)\), tending to zero as \(u,v\to0\), are monotone for small values of the arguments. By \(f^{-1}(v)\) is denoted the function inverse to the function \(f(u)\).
If \(Q(u)\ll u\) and \(u<Q(u)\), then the class of spaces \(l_{M_1}\) satisfying the condition of Theorem 3 is nonempty. In the first case \(Q(u)\) is equivalent to some \(N\)-function; in the second case the relation \(M_1(u)<Q(u)\) is valid for any \(N\)-function \(M_1(u)\).
If conditions (1) guarantee the continuity of the operator \(A\) as an operator acting from the space \(l_M\) into some coordinate space contained in the space \(m\) of bounded sequences, then the following relation is satisfied: \(N(v)<P(v)\), where \(N(v)\) is the function complementary to \(M(u)\). From Young’s inequality it follows:
Theorem 5. Let the first of conditions (1) be satisfied and let \(N(v)\sim P(v)\). Then the operator (2)—(3) acts continuously from the space \(l_M\) into the space \(m\).
Below it is assumed that \(N(v)\ll P(v)\). Then the function \(R^{-1}(v)=v/N^{-1}[P(v)]\) is monotone for small \(v\), and its inverse function \(R(u)\) is equivalent to some \(N\)-function.
Theorem 6. Let \(N(v)\ll P(v)\). Let the function \(Q[R(u)]\) be equivalent to some \(N\)-function \(M_1(u)\) satisfying, for small \(u\), the \(\Delta_2\)-condition and the \(\Delta_1\)-condition:
\[
M_1(\alpha u v) \leq M_1(u)M_1(v)\quad (u,v\leq u_0).
\]
If \(M_1(u)\sim M(u)\) with \(M_1(u)\ll M(u)\), then the operator (2)—(3) acts continuously from the space \(l_M\) into any space \(l_{M_2}\) for which \(M_2(v)\) satisfies inequality (4).
If the function \(N_1(v)\), complementary to \(M_1(u)\), also satisfies the \(\Delta_1\)-condition for all \(u, v\), then the operator \(A\) acts continuously from \(l_M\) into \(l_{M_1}\).
If the functions \(P(u)\) and \(Q(u)\) are equivalent and, then the function \(Q[R(u)]\) is equivalent to the \(N\)-function \(M(u)\).
Theorem 7. If the series \(\sum_{i=1}^{\infty} Q(a_{ik})\) converge uniformly with respect to \(k\), then under the conditions of Theorem 6 the operator \(A\) acts from \(l_M\) into \(h_{M_2}\) (or into \(h_M\)) and is completely continuous.
It follows from Theorems 3, 5 and 6, in the particular case (without estimating the norm of the operator), the following theorem of Hardy, Littlewood and Pólya (\(^6\), p. 239):
Let \(r,t>0\),
\[ \sum_{k=1}^{\infty} |a_{ik}|^r \leq C_1,\qquad \sum_{i=1}^{\infty} |a_{ik}|^t \leq C_2. \tag{6} \]
Let \(p \geq 1\) and \(p'=\dfrac{p}{p-1}\geq r\). Let
\[ t \geq \left(1-\frac{r}{p'}\right)p. \tag{7} \]
Then the operator (2)—(3) acts continuously from \(l_p\) into \(l_{p_1}\), where
\[ p_1=\frac{t}{1-r/p'} \quad (l_\infty=m,\ \text{if } p'=r). \]
This theorem is naturally supplemented by the following assertion:
Theorem 8. If \(p'>r\) and the series \(\sum_{i=1}^{\infty} |a_{ik}|^t\) converge uniformly with respect to \(k\), then under the conditions of the preceding theorem the operator \(A\) is completely continuous as an operator from \(l_p\) into \(l_{p_1}\).
The conditions (6) are less restrictive than the usual sufficient conditions for complete continuity of matrix operators in the spaces \(l_p\) (\(p>1\)) (see, for example, (\(^7\), pp. 322—323). Thus, for example, from the well-known Koch condition
\[ \sum_{k=1}^{\infty}\sum_{i=1}^{\infty} |a_{ik}|^2<\infty \]
one cannot obtain an assertion on complete continuity in \(l_2\) of the diagonal operator
\[ A= \left| \begin{array}{ccccccc} \alpha_1 & 0 & 0 & . & . & . & .\\ . & . & . & . & . & . & .\\ . & . & . & . & . & . & .\\ 0 & 0 & . & . & \alpha_n & . & .\\ . & . & . & . & . & . & . \end{array} \right|, \]
where \(\alpha_n\to 0\) and \(\sum_{n=1}^{\infty}\alpha_n^2=\infty\). At the same time, the complete continuity of such an operator follows from Theorem 8.
Voronezh Civil Engineering Institute
Received
23 III 1962
REFERENCES
\(^1\) W. Orlicz, Bull. Intern. de l’Acad. Polon., Ser. A, No. 3—4 (1936).
\(^2\) Yu. I. Gribanov, Proceedings of the Seminar on Functional Analysis, Voronezh State Univ., issue 6 (1958).
\(^3\) Yu. I. Gribanov, Izv. Vyssh. Uchebn. Zaved., Mathematics, No. 4 (1958).
\(^4\) T. Ando, Math. Ann., 140, 174 (1960).
\(^5\) S. Banach, Course of Functional Analysis, Kiev, 1948.
\(^6\) G. G. Hardy, J. E. Littlewood, G. Pólya, Inequalities, IL, 1948.
\(^7\) L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, 1959.