Reports of the Academy of Sciences of the USSR
1970. Volume 193, No. 4

UDC 513.881

MATHEMATICS

M. M. DRAGILEV

ON MULTIPLE REGULAR BASES IN KÖTHE SPACE

(Presented by Academician L. V. Kantorovich on 26 IX 1969)

1. Let \(E\) be a Köthe space*; let \((x_n)\) be an arbitrary absolute basis of it; let \((y_n)\) be one of those absolute bases in \(E\) which can be obtained as a result of the following three operations: a) the mapping \((x_n)\to (Tx_n)\), where \(T\) is an isomorphism of the space \(E\) onto itself; b) multiplication of the elements \(Tx_n\) by numbers \(\lambda_n\), \(\lambda_n\ne 0\) \((n=1,\ldots)\); c) permutation of the terms of the sequence \((\lambda_n Tx_n)\). The bases \((x_n)\) and \((y_n)\) are called, respectively, equivalent, pre-equivalent, and quasi-equivalent. Up to the present time the question has not been completely solved (see, for example, \((^{1})\)): in what case are all absolute bases of a space in one of the listed equivalence relations?

It is not difficult to see that any two bases in \(E\) are equivalent if and only if the space is finite-dimensional (here it is essential that the topology in \(E\) can be given by a countable set of norms \((^{3})\)). Analogously, all absolute bases in \(E\) are pre-equivalent if and only if \(E\) is normable (i.e., degenerates into the Banach space \(l_1\))**. It is not known, however, whether all absolute bases of a nondegenerate space \(E\) are quasi-equivalent. A positive answer has been obtained only for some special classes of nuclear and Montel Köthe spaces possessing a regular*** basis \((^{3-9})\).

In the present article a more general case is considered. Let \((x_n)\) be an arbitrary sequence in \(E\); let \(\mu\) be the minimal number of its regular subsequences such that each element \(x_n\) is contained in one and only one of them. Put \(m=\inf \mu\), where the infimum is taken over the set of all possible permutations of the elements \(x_n\) \((n=1,2,\ldots)\). A sequence for which \(m\ne 1\) cannot be made regular by a permutation of its elements. It is natural to call it an \(m\)-fold regular sequence if \(m\) is finite and \(\mu=m\). We shall extend this definition also to the case when \(m=1\) or \(m=\infty\). As will be shown, the multiplicity \(m\) of an arbitrary basis in a nuclear space \(E\) depends only on the space. In other words, each nuclear Köthe space belongs to one and only one of the classes: \(\mathcal N_s=\{E:m=s\}\), \(1\le s\le\infty\). All the classes \(\mathcal N_s\) are nonempty. Any of them, with the exception, perhaps, of \(\mathcal N_\infty\), contains spaces all of whose bases are quasi-equivalent.

1. Let \((x_n)\) be a sequence in \(E\), the elements of which are not equal to zero; let \((t_n)=t\) be an arbitrary numerical sequence and

\[ |t|_p=\sum_n |t_n|\,\|x_n\|_p \qquad (p=1,2,\ldots). \]

We agree to denote by \([x_n]\) the Köthe space

\[ \{t:\ |t|_p<\infty,\ p=1,\ldots\} \]

(with the topology given by the system—

* A Köthe space is a complete countably normed space with an absolute Schauder basis.

** A more particular assertion is proved in \((^{10})\).

*** A sequence \((x_n)\), \(x_n\in E\) \((1\le n<\nu\le\infty)\), is called regular if there exists a defining system of norms in \(E\) such that, for any \(p,q=1,2,\ldots\), the sequence of numbers \(\|x_n\|_p/\|x_n\|_q\) \((1\le n<\nu\le\infty)\) is monotone \((^{7})\).

by the system of norms \(|\cdot|_p\). If \(E\) is a Montel space with an absolute regular basis \((x_n)\), then denote

\[ K(E)=\{[\lambda_n x_{k_n}]:\ k_n\to\infty,\ \lambda_n>0,\ n=1,\ldots\}. \]

Suppose that all regular bases in \(E\) are pre-equivalent (in this case \(K(E)\) does not depend on the choice of basis) and, moreover, that every space contained in \(K(E)\) has the analogous property. The class \(\mathscr E\) of all spaces \(E\) under consideration is nonempty (it contains, in particular, spaces of type \((d_i)\), \(i=1,2\) \((^7)\)). Moreover, \(E\in\mathscr E\) if and only if \(K(E)\subset\mathscr E\).

For arbitrary spaces \(E_i\in\mathscr E\) \((i=1,2)\), put

\[ K(E_1)\oplus K(E_2)=\{G_1\oplus G_2:\ G_i\in K(E_i),\ i=1,2\}. \]

Let \(R\) be a subset of the class \(\mathscr E\), closed with respect to the operation \(\oplus\). We say that the function \(K(E)\) is additive on \(R\), if \(K(E_1\oplus E_2)=K(E_1)\oplus K(E_2)\) for any \(E_i\in R\) \((i=1,2)\).

Definition 1. A maximal subset \(R\subset\mathscr E\), closed with respect to the operation \(\oplus\), will be called a Riss class if the function \(K(E)\) is additive on \(R\).

For example, the set \(R_0\) (respectively, \(R_\infty\)) of all Köthe spaces that are finite (infinite) centers of Riss scales \((^5)\) is a Riss class. Let, in general, \(E\) be an arbitrary space of type \((d_1)\) or \((d_2)\). The following holds.

Theorem 1. The set \(K(E)\) is a Riss class if and only if \(E\) is isomorphic to each of its subspaces of finite codimension.

In what follows we call a Riss class complete if there exists a space \(E\in R\) such that \(K(E)=R\).

Let us note some corollaries. Let \(f(u)\) be a nondecreasing odd function of a real argument, logarithmically convex for \(u\ge 0\). In \((^7)\) a broad class of Montel Köthe spaces \(L_f(b,r)=[(\delta_{nj})_{j=1}^{\infty}]\) is considered, for which

\[ |(\delta_{nj})_{j=1}^{\infty}|_p=\exp f(r_p b_n)\quad (r_p\uparrow r), \]

where \(-\infty<r\le\infty\), \(b=(b_n)\), and \(b_n\uparrow\infty\). By definition,

\[ (f)_\sigma=\{L_f(b,r):\ b_n\uparrow\infty,\ r=\sigma\}\quad (\sigma=-1,0,1,\infty). \]

There, countable families of pairwise disjoint classes \((f)_\sigma\) are also singled out.

Corollary 1. The set \((f)_\sigma\) is a complete Riss class.

Corollary 2. The cardinality of a maximal set of pairwise disjoint complete Riss classes is at least \(c\).

Corollary 3. The set \(\mathfrak R\) of all Riss classes is uncountable.

We formulate necessary and sufficient conditions for the classes \(R_i\in\mathfrak R\) \((i=1,2)\) not to intersect. Let \(T:E_1\to E_2\) be a continuous linear operator that carries an absolute basis \((x_n)\) of the space \(E_1\in R_1\) into an absolute basis of the space \(E_2\in R_2\).

Theorem 2. The following assertions are equivalent:

1°. \(R_1\cap R_2=\varnothing\).

2°. Whatever the spaces \(E_i\in R_i\) \((i=1,2)\), the absolute basis \((x_n)\) in \(E_1\), and the operator \(T\), there exists a subsequence \((x_{j_n},\, n=1,2,\ldots)\) such that the restriction of \(T\) to the corresponding subspace

\[ \operatorname{span}(x_{j_n})\subset E_1 \]

is compact.*

If the classes \(R_1\) and \(R_2\) are such that all continuous linear mappings of each space \(E_1\in R_1\) into any space \(E_2\in R_2\) are compact, then we shall write \(R_1>R_2\).

Definition 2. Classes \(R_i\in\mathfrak R\) \((i=1,2)\) will be called essentially distinct if \(R_1>R_2\) or \(R_2>R_1\).

* For the definition of a compact operator, see, for example, \((^{11})\).

For example, the classes \(R_0\) and \(R_\infty\) are essentially different, and moreover \(R_0 > R_\infty\) \((^2)\). As V. P. Zaharyuta showed (ibid.), there exist countable families of Riss classes \((f)_\sigma\), linearly ordered by the relation \((\geq)\). It is not known whether any nonintersecting classes \(R_i \in \mathfrak R\) \((i=1,2)\) are essentially different.

Remark. Nonintersecting subsets \(K(E_i) \subset R_i\) \((i=1,2)\) cannot be essentially different in the sense of Definition 2.

Let, further, \(E=\bigoplus E_i\), where \(E_i \in R_i \in \mathfrak R\) \((i=1,\ldots,k)\), and \(R_i=K(E_i)\) for \(i \leq s \leq k\), i.e. all the Riss classes \(R_1,\ldots,R_s\) are complete, and the corresponding spaces \(E_i\) satisfy the condition of Theorem 1. Important for what follows is

Theorem 3. If \(1 \leq s \leq k\) and all \(R_i\) are distinct (respectively, \(1 \leq s < k\) and \(R_i \cap R_j=\varnothing\) for \(i \leq s < j\)), then in the space \(E\) there exists an \(m\)-fold regular basis, for which \(m \geq s\) (respectively \(m \geq s+1\)). Moreover, equality is attained in every case.

1. Let \(E \subset \mathcal N\), where \(\mathcal N\) is the class of all nuclear Köthe spaces. As is known \((^{12})\), all bases in \(E\) are absolute.

Lemma (cf. \((^7,^8)\)). Whatever the bases \((x_n)\) and \((y_n)\) in \(E\), there exist two sequences \((k_n)\), \(k \to \infty\), and \((\lambda_n)\), \(\lambda_n>0\) \((n=1,2,\ldots)\), such that \([y_n]=[\lambda_n x_{k_n}]\).

We give a number of corollaries, some of which are of independent interest.

Corollary 1. In a nuclear space with a regular basis, the multiplicity \(m\) of an arbitrary basis is equal to one.

Corollary 2. In a nuclear space \(E \in \mathscr E\), all bases are quasiequivalent.

Extend to the whole class \(\mathcal N\) the function \(K(E)\) defined on the set \(\mathcal N \cap \mathscr E\) (here, as \((x_n)\), an arbitrary basis of the space is taken).

Corollary 3. The function \(K(E)\) does not depend on the choice of a basis in \(E\) and is a topological invariant on the class \(\mathcal N\).

Corollary 4. The multiplicity \(m\) of an arbitrary basis in a nuclear space \(E\) is a constant depending only on \(E\).

Corollary 5. Every space \(E \in \mathcal N\) belongs to one and only one of the classes \(\mathcal N_s\) \((1 \leq s \leq \infty)\).

Hence, and also from Theorems 1 (Corollary 2) and 3, it follows that

Corollary 6. None of the classes \(\mathcal N_s\) \((1 \leq s \leq \infty)\) is empty.

Let, further, \(E^j\) denote the subspace of the space \(E\) of codimension \(j\) for \(j \geq 0\), and the direct sum of the form \(E \oplus L\), where \(\dim L=-j\), for \(j<0\). From Corollary 3 there follows a theorem of conditional type.

Theorem 4. Let \(E=\bigoplus E_i\), where \(E_i \in R_i\) \((i=1,\ldots,s)\) are nuclear spaces, and \(R_i \cap R_k=\varnothing\) if \(i \ne k\). The following assertions are equivalent:

\(1^\circ\). In the space \(E\) all bases are quasiequivalent.

\(2^\circ\). Whatever the nuclear spaces \(G_i \in R_i\) \((i=1,2,\ldots,s)\), the isomorphism \(E \sim \bigoplus G_i\) holds if and only if there are \(j_i\), \(\sum_i j_i=0\), such that \(E_i \sim G_i^{j_i}\) \((i=1,\ldots,s)\).

It remains for us to apply the following result.

Theorem 5 (see \((^2)\)). Let \(X_i\) and \(Y_i\) \((i=1,2)\) be linear topological spaces such that every continuous linear mapping from \(X_1\) to \(Y_2\) and from \(Y_1\) to \(X_2\) is compact. The products \(X_1 \times X_2\) and \(Y_1 \times Y_2\) are isomorphic if and only if, for some finite \(j\), \(X_1 \sim Y_1^j\) and \(X_2 \sim Y_2^{-j}\).

An immediate consequence of Theorems 3, 4, and 5 is

Theorem 6*. In the space \(E=\bigoplus E_i\), where \(E_i \in R_i\) \((i=1,2,\ldots,s)\)—

* This theorem was proved by me jointly with V. P. Zaharyuta.

nuclear Köthe spaces belonging to essentially different (ordered) Riss classes, all bases are quasi-equivalent. Moreover \(E \in \mathcal N_s\), if \(R_i = K(E_i)\) \((i = 1, \ldots, s)\).

Rostov State
University

Received
22 IX 1969

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