V. L. Levin

FUNCTORS IN CATEGORIES OF BANACH SPACES DEFINED BY (KB)-LINEALS

(Presented by Academician L. V. Kantorovich on 23 XI 1964)

Functors in categories of Banach spaces were considered in the papers of A. S. Shvarts ((^1)) and B. S. Mityagin and A. S. Shvarts ((^2)). In these works a theory of duality of functors in the category of Banach spaces was constructed; in particular, functors defined by normed ideals of sequences and of measurable functions were studied.

In the present note, on the tensor product (E \otimes X) of a (KB)-lineal (E) and a Banach space (X), a natural crossnorm is defined, and the functor (\Phi_E) is studied, assigning to each (X) the completion of (E \otimes X) with respect to this crossnorm. The functor (\Phi_E) includes, as special cases, functors defined by minimal normed ideals of sequences and of measurable functions, as well as other concrete functors. A realization of the space (\Phi_E(X')), the space of mappings ({\Phi_E \to \Phi_F}), and the dual functor (D\Phi_E) are described. As consequences, answers are obtained to some questions posed in ((^2)), and a theorem on the nuclearity of one class of composite mappings.

A crossnorm on the tensor product (X \otimes Y) of Banach spaces (X,Y) is a norm (p) satisfying the conditions: (p(x \otimes y)=|x|X|y|_Y) for all (x \in X,\ y \in Y), and
(p'(x' \otimes y')=|x'|) for all (x' \in X',\ y' \in Y'), where (p') is the norm on ((X \otimes_p Y)') conjugate to (p), and (x' \otimes y') is the functional on (X \otimes_p Y) acting by the formula}|y'|_{Y'

[
\left\langle \sum_{k=1}^{n} x_k \otimes y_k,\ x' \otimes y' \right\rangle
=
\sum_{k=1}^{n} \langle x_k,x'\rangle \langle y_k,y'\rangle .
\tag{3}
]

Among crossnorms there exist the maximal (\pi) and the minimal (\varepsilon) ((^3,^4)). The completions of (X \otimes Y) with respect to the norms (\pi) and (\varepsilon) are denoted by (X \widehat{\otimes} Y) and (X \overset{\vee}{\otimes} Y).

Let (\mathcal F) be a functor of type (\Sigma) in some category (\mathcal K) of Banach spaces. It can be shown that for every (X \in \mathcal K), (\mathcal F(X)) is isometric to the completion of (\mathcal F(I)\otimes X) with respect to some crossnorm, and for (\alpha \in (X \to Y))
[
\mathcal F(\alpha)=1_{\mathcal F(I)}\otimes \alpha
]
and for (\varphi \in {\mathcal F \to \mathcal G}), where (\mathcal G) is a functor in (\mathcal K),
[
\varphi_X=\varphi_I \otimes 1_X .
]
The identity mapping of (\mathcal F(I)\otimes X) onto itself extends to a continuous linear mapping of (\mathcal F(X)) into (\mathcal F(I)\widehat{\otimes}X); we shall say that the functor (\mathcal F) satisfies the condition of mutual uniqueness* if this mapping is one-to-one for every (X \in \mathcal K).

Let (E) be a (KB)-lineal ((^5)), and (X) a Banach space. Introduce on (E\otimes X) the norm (n_E), putting, for (z=\sum_{k=1}^{n} e_k\otimes x_k),

[
n_E(z)=
\inf\left{
|u|E:\
u \ge
\left|
\sum e_k\langle x_k,x'\rangle}^{n
\right|
\text{ for all } x',\ |x'|\le 1
\right}.
]

It is verified directly that the function (\Phi_E), assigning to each (X \in \mathcal K) the completion of (E\otimes X) with respect to the norm (n_E), and to each (\alpha \in (X\to Y)) the mapping
[
\Phi_E(\alpha)=1_E\otimes \alpha,
]
is a functor in any category (\mathcal K).

Theorem 1**. If (E) is, respectively, the minimal normed ideal of sequences (n), the corresponding ideal of measurable func-

* Concerning the definitions and notation relating to functors in categories of Banach spaces, see ((^1,^2)).

** For definitions of normed ideals of sequences and measurable functions and of the functors defined by them, as well as of the functor (C), see ((^2)).

… (N), the space (C) of continuous functions on the segment ([a,b]), then the functor (\Phi_E) is, respectively, (n, N, C).

Proof. In the cases under consideration there exists

[
u_0=\sup_{|x'|\leqslant 1}\left|\sum_{k=1}^{n} e_k\langle x_k,x'\rangle\right|\in E
\quad \text{for every } \quad
z=\sum_{k=1}^{n} e_k\otimes x_k
\quad \text{and} \quad
n_E(z)=
]

[
=|u_0|_E.
]

Further, since the supremum in a (K B)-lineal of sequences (functions) is determined coordinatewise (pointwise), the norm (n_E) coincides on (E\otimes X) with the norm of the corresponding space (E(X)), and the assertion of the theorem follows from the fact that (E\otimes X) is dense in (E(X)) ((E=n,N,C)).

A mapping (a\in (X\to E)) that carries the unit ball of (X) into a set (E) bounded in the sense of the ordering (see ((^5))) will be called proper*. On the space of proper mappings we introduce the norm

[
n(a)=\inf{|u|_E:u\geqslant |ax| \quad \text{for all } x,\ |x|\leqslant 1}.
]

We shall say that a (K B)-lineal (E) satisfies condition (()) if, for every monotonically increasing sequence (0\leqslant e_n\in E) unbounded above, (|e_n|\to\infty) as (n\to\infty). All (K B)-spaces ((^5)) (in particular (l^p) and (L^p)) and the space (C) satisfy condition (()).

Theorem 2. If (E) satisfies condition (()), and (\Phi_E) satisfies the condition of mutual uniqueness, then (\Phi_E(X')) is isometric to the closure of the set of continuous finite-dimensional mappings of (X) into (E) in the space of proper mappings of (X) into (E). If, moreover, there exists a sequence of finite-dimensional operators (v_n\in (E\to E)) such that (v_n e\to e) as (n\to\infty) and (v_n e\leqslant e) for all (0\leqslant e\in E), then (\Phi_E(X')) is isometric to the space of all proper mappings of (X) into (E).*

Remark 1. (\Phi_E) satisfies the condition of mutual uniqueness for (E=n,N,C).

Remark 2. A sequence (v_n) with the required properties exists for every space of sequences with the usual ordering, in which the “orts” (\delta_n=(0,\ldots,0,\underset{n}{1},0,\ldots)) form a basis.

Let us outline the proof. According to the condition of mutual uniqueness, each element (z\in \Phi_E(X')) may be regarded as a completely continuous mapping of (X) into (E). Using condition ((*)), one can show that this mapping is proper. Since the norm (n_E) of the element (z=\sum_{k=1}^{n} e_k\otimes x_k') is equal to the norm (n) of the corresponding mapping (\sum_{k=1}^{n}\langle\,\cdot\,,x_k'\rangle e_k), the first part of the theorem follows. Let (a) be a proper mapping of (X) into (E) and (|ax|\leqslant e_0) for all (x\in X,\ |x|\leqslant 1). Then (|(1_E-v_n)ax|\leqslant (1_E-v_n)e_0) for all (x\in X,\ |x|\leqslant 1), and

[
n(a-v_na)=\inf{|u|:u\geqslant |(1_E-v_n)ax| \text{ for all } x;\ |x|\leqslant 1}\leqslant |(1_E-v_n)e_0|,
]

so that (a) is approximated in the norm (n) by finite-dimensional mappings (v_n a), and, consequently, (a\in\Phi_E(X')).

Let (E,F) be (K B)-lineals. A mapping (a\in(E\to F)) is called regular if it can be represented as the difference of two positive mappings (see ((^5))).

We shall say that a (K B)-lineal (E) satisfies condition ((**)) if, for every monotonically increasing sequence (0\leqslant e_n\in E) possessing a supremum (\sup_n e_n),

[
|e_n|\to \left|\sup_n e_n\right|
]

as (n\to\infty). All (K B)-spaces, normalized ideals of sequences and measurable functions, and the space (C) satisfy condition ((**)).

* When (E) is an (\mathcal K^+)-space, proper mappings coincide with bounded mappings (see ((^5)), p. 246).

Theorem 3. Let (E) be a (KB)-lineal, and let (F) be a conditionally complete (KB)-lineal (see ((^5))), satisfying conditions (()) and ((*)), and suppose that either (E) or (F) is separable. Then the space of mappings ({\Phi_E\to\Phi_F}) is isometric to the Banach space ((E\to F)r) of regular mappings of (E) into (F) with norm
(\nu(\alpha)=||\alpha||); under this isometry the mapping
(\alpha\in(E\to F)_r) corresponds to the class of mappings
(\alpha\otimes 1_X\in(\Phi_E(X)\to\Phi_F(X))), (X\in\mathfrak K). The assertion of the theorem is valid in every category (\mathfrak K) containing the space (c_0).

Remark 1. Theorem 3 remains valid if, instead of separability of (E) or (F), one requires that every set in (E) bounded in the order sense be separable.

Remark 2. For the case (E=F=L^p), Theorem 3 gives a positive answer to the question about the ring of operators in the functor (L^p), posed in ((^2)).

Lemma. Let (E,F) be (KB)-lineals. To a regular mapping (\alpha:E\to F) there corresponds a mapping of functors (\Phi_E\to\Phi_F) in every category (\mathfrak K), and
[
|\alpha|_{{\Phi_E\to\Phi_F}}\leq \nu(\alpha).
]

Proof of Theorem 3. Suppose (\alpha\in(E\to F)) is not regular. Then in (E) there is a set (Q), bounded in the order sense, such that (\alpha Q) is not bounded in the same sense (see ((^5)), p. 232). Take a sequence of elements (e_n\in Q) whose images (\alpha e_n) are dense in (\alpha Q). Then the sequence
[
g_n=\sup_{1\leq k\leq n}|\alpha e_k|
]
is not bounded in (F) in the order sense; applying condition ((*)), we obtain that (|g_n|\to\infty) as (n\to\infty). Consider the elements
[
z_n=\sum_{k=1}^n e_k\otimes\delta_k\in\Phi_E(c_0),
]
where
[
\delta_k=(0,\ldots,0,1,0,\ldots).
]
We have
[
|z_n|{\Phi_E(c_0)}
=
\left|\sup|e_k|\right|E
\leq |e_0|_E,
]
where (e_0) is an element of (E) majorizing the moduli of the elements of (Q). On the other hand,
[
|(\alpha\otimes 1)z_n|{\Phi_F(c_0)}
=
\left|\sup|\alpha e_k|\right|_F
=
|g_n|_F\to\infty
]
as (n\to\infty), so that
[
\alpha\notin{\Phi_E\to\Phi_F}
]
if (\mathfrak K\ni c_0). Hence, and from the lemma, it follows that
[
(E\to F)_r={\Phi_E\to\Phi_F}.
]

Now let (\alpha\in(E\to F)r). Fix (0\leq e_0\in E) and take a sequence of elements
[
e_n\in Q={e: |e|\leq e_0},
]
whose images (\alpha e_n) are dense in (\alpha Q_{e_0}). Put
[
u_n=\sup_{1\leq k\leq n}|\alpha e_k|.
]
Then
[
\sup_{|e|\leq e_0}|\alpha e|=\sup_n u_n,
]
and, by ((**)),
[
\left|\sup_{|e|\leq e_0}|\alpha e|\right|
=
\lim_{n\to\infty}|u_n|.
]
Moreover,
[
|u_n|F
=
\left|\sum^n \alpha e_k\otimes\delta_k\right|{\Phi_F(c_0)}
\leq
|\alpha|
\left|\sum_{k=1}^n e_k\otimes\delta_k\right|{\Phi_E(c_0)}
]
[
=
|\alpha|
\sup_{1\leq k\leq n}|e_k|
\leq
|\alpha|{{\Phi_E\to\Phi_F}}|e_0|.
]
Passing to the limit as (n\to\infty), we obtain
[
\left|\sup|\alpha e|\right|
\leq
|\alpha|{{\Phi_E\to\Phi_F}}|e_0|.
]
Further, since
[
|\alpha|e_0=\sup|\alpha e|
]
(see ((^5)), p. 231),
[
\nu(\alpha)
=
\sup_{0\leq e_0\in E}\frac{||\alpha|e_0|}{|e_0|}
=
\sup_{0\leq e_0\in E}
\frac{\left|\sup_{|e|\leq e_0}|\alpha e|\right|}{|e_0|}
\leq
|\alpha|{{\Phi_E\to\Phi_F}}.
]
Thus
[
\nu(\alpha)=|\alpha|,
]
and the theorem is proved.

We shall say that a (KB)-lineal (E) satisfies condition (())* if, for every Banach space (X) and every (z\in E\otimes X),
[
n_E(z)=\inf\left|\sum_{k=1}^n |e_k|\,|x_k|\right|,
]
where the infimum is taken over all representations
[
z=\sum_{k=1}^n e_k\otimes x_k.
]
It can be shown that normed ideals

spaces of sequences and measurable functions, and the space (C), satisfy condition (( * )); this condition is satisfied by every (KB)-lineal of functions (E) with the usual ordering, containing the step functions as a dense subspace.

Call a mapping (\alpha:E\to X) summing if it carries every series (\sum_{k=1}^{\infty} e_k) in (E) for which the series (\sum_{k=1}^{\infty}|e_k|) converges into an absolutely convergent series(^*) (\sum_{k=1}^{\infty}\alpha e_k) in (X). On the space of summing mappings consider the norm
[
v_E(\alpha)=\sup \sum_{k=1}^{n}|\alpha e_k|X\bigg/\left|\sum|e_k|\right|_E,}^{n
]
where the supremum is taken over all possible finite sets of elements (e_k\in E), (k=1,\ldots,n).

Theorem 4. Let (E) be a (KB)-lineal satisfying condition (( * )). Then (D\Phi_E(X)) is isometric to the space of summing mappings of (E) into (X) in every category (\mathfrak K) containing the space (l^1).

Remark. In ((^2)) this theorem was proved for (E=C).

Let (B(E)) be the space of sequences ((e_k)\subset E) for which the series (\sum_{k=1}^{\infty}|e_k|) converges, with norm
[
|(e_k)|=\left|\sum_{k=1}^{\infty}|e_k|\right|_E.
]

Lemma 1. (B(E)) is complete for every (KB)-lineal (E).

Let (\alpha) be a summing mapping from (E) into (X). Consider the mapping (\Psi_\alpha:B(E)\to l^1(X)), defined by the formula (\Psi_\alpha(e_k)=(\alpha e_k)).

Lemma 2. (\Psi_\alpha) is continuous for every (KB)-lineal (E).

Lemma 3. Let (E) be a (KB)-lineal satisfying condition (( * )), and let (\alpha) be a summing mapping from (E) into (X). Then (\alpha) generates an element of (D\Phi_E(X)) in every category (\mathfrak K), and
[
|\alpha|_{D\Phi_E(X)}\leq v_E(\alpha).
]

Lemma 4. Let (E) be a (KB)-lineal and let (\alpha:E\to X) generate an element of (D\Phi_E(X)) in a category (\mathfrak K) containing (l^1). Then (\alpha) is a summing mapping and
[
v_E(\alpha)\leq |\alpha|_{D\Phi_E(X)}.
]

Theorem 4 follows from Lemmas 3 and 4; Lemma 3 rests on Lemmas 1 and 2.

In ((^2)) the problem was posed of describing the functor (DN) in the category of all Banach spaces, and the conjecture was made that (DN=N').** This conjecture is not true, since the embedding (N\subset L^1) determines an element of (DN(L^1)) not belonging to (N'(L^1)). For the case of a minimally normed ideal, Theorem 4 gives a description of the functor (DN).

Theorem 5. Let (X,Y) be Banach spaces, (E) a (KB)-lineal satisfying conditions (()) and (( * *)), (\Phi_E) satisfy the condition of mutual one-to-one correspondence, (u:X\to E) a proper mapping approximable by finite-dimensional mappings in norm (n), and (\alpha:E\to Y) a summing mapping. Then their product (\alpha u) is a nuclear mapping from (X) into (Y).

Received
16 XI 1964

REFERENCES

(^1) A. C. Schwartz, DAN, 149, No. 1, 44 (1963). (^2) B. S. Mityagin, A. C. Schwartz, UMN, 19, issue 2 (116), 65 (1964). (^3) R. Schatten, A theory of Cross-Spaces, Princeton, 1950. (^4) A. Grothendieck, Mem. Am. Math. Soc., 16 (1955). (^5) B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Moscow, 1961.

(^) A series (\sum_{k=1}^{\infty}x_k) in a Banach space (X) is called absolutely convergent* if
[
\sum_{k=1}^{\infty}|x_k|<\infty.
]

(^ {**}) (N') denotes the associated ideal ((^2)).