Some equivalent classes oî Kothe spacesAbstract.

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PEACE MATEMATYCZNE X X (1978)

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Some equivalent classes oî Kothe spaces

A b s tr a c t . The following result is obtained.

T

. Let E be a Kothe space. The following are equivalent:

(i) E is regular and D x (resp. D 2).

(ii) E is pseudo-regular and D x resp. D2).

(iii) E is a Goo (resp. Gx) space.

In the past few years several special classes of Kothe spaces have arisen in the study of nuclearity, both in the theory of A-nuclearity and in the basic structure theory, (cf. [1]-[11]). I t is not surprising th a t in such a developing theory many of the concepts introduced by different authors are very similar in nature. I t is the purpose of this note to show th a t some of these definitions are equivalent.

Let (a*) be a m atrix of numbers such th a t 0 < < a j+1. The Kothe space E = E(a„) is the sequence space E = {t: Vic, № = i X l « î <

The Kothe dual of E is E x = {t: З&э |#n| < Mafy. The Kothe space E is regular [3] if E = E(a„), where

,fc+i ,* + l

Ьи-i for all k, n. /

The Kothe space E is pseudo-regular [2] if E = E{a„), where Vfc3 p э Vm3$ э Vrt

a'

3<n Щ

lb ^ .

— < + o o .

ai

The Kothe space E is (Bx) [ 1 ] if E = E(a*), where for all n, a\ = 1 , and for all k, there exists m such th a t (a *)2 < a™ for all n.

The Kothe space E is (D2) [ 1 ] if E — E(a^), where for all к and n ^ k , a% = 1 , and for all к there exists p such th a t for all

The Kothe space E is a O^-space [9.] if E — Е(а*), where

(i) a* = 1 for all n.

450 W. Rob inson

(ii) ak < a *+1 for all k, n.

(iii) For all к there exists p such th a t

{akf ^ an f°r all »•

The Kothe space E is a Gx-space [9] if E = E(a*), where (i) For all к there exists p such th a t ak < (ap)2 for all n.

(ii) ak > a%+1 for all к, n.

L emma , (i) Let E be (L>x) and pseudo-regular. Then there is a matrico {bk) f or E su°b, that for all k, n, bk < b^+ l.

(ii) Let E be (D2) and pseudo-regular. Then there is a matrico (bk) for E such that for all k, n, bk > bk+1.

P ro o f, (i) Let E = E(ak). The diametral dimension A(E) — ( E ‘E X)X [7]. Since E is (_DX), (E -E x)x = E x [ 8 ]. Given a e E x, let bn = sup a,.

j^n Then, bn > bn+1 for all n. By Lemma 4.13 of [7], b e E x. Hence, for all к there exists bk e E x such th a t ak < bk for all n and bk < bk+1 for all k, n.

I t is easy to see th a t E(ak) = E(bk) so (bk) is the desired matrix.

The proof of (ii) is similar.

T

1 . Let E be a Kothe space. The following are equivalent:

(a) E is (Df) and regular.

(b) E is (D x) and pseudo-regular.

(c) E is a G^-space.

P ro o f. (a)->“(b) follows from [2], Proposition 1 . (b) ->(c) follows easily from Lemma (i).

(c) ->(b) let E{ak) be a G^-space.

к l)k + 1

Let bk = J] ajn. Then ~ t ~ = uk which is increasing and the space E

i bn

is regular. Also bln = 1 for all n and for all к 3 p x, . . . , p k such th a t for all n

к к

(bkf — {П « I )2 < П K J ^ bPk. Hence, E{bk) is (Dx) and regular, so we

j=k j =1

need to check th a t E (ak) = E(bk). But clearly ak ^ b k, and by repeated application of property (iii) of the Gœ space E(ak) we see th a t for all к there exists p such th a t bk < a* for all n. Hence, E(ak) — E(bk).

T

2. Let E be a Kothe space. The following are equivalent.

(a) E is (T>2) and regular.

(b) E is (D2) and pseudo-regular.

(c) E is a Gx-space.

P ro o f. (a)-^(b)->(c) as in Theorem 1.

(c)^ (a) : Let E (ak) be a 6 rr space. W ithout loss of generality we

may assume th a t ak — 1 if к > n, and th a t for all к, n, ak < (ak+lf . Then

Kôthe spaces 461

we can define bk = f [ a 3 n if n > ft and bk = 1 if n < Tc. Then it is easy to j=k

see th a t E(bk) is regular and (D2). To see th a t E(ak) = E(bk) observe th at for all ft, bk = ak if n < ft and b\ < ak if n > ft, since all ajn > 1 . On the other hand, if n < ft

_

Vn Orn ,tt— ¹ « î x c i ’ r 1 > (<)"' > a% к - 1 Thus, for all ft, < bk, so Я(л*) = E(bkn).

R e f e r e n c e s

[1] C. B e s s a g a , Some remarks on Dragilev's theorem, Studia Math. 31 (1968), p. 307-318.

[2] L. C rone, Ed D u b in s k y and W. B. R o b in s o n , On pseudo-regular bases in power series spaces, Journal of Functional Analysis (to appear).

[3] M. M. D r a g ile v , On regular bases in nuclear spaces, Amer. Math. Soc. Trans.

93 (1970), p. 61-82.

[4] — Kôthe spaces differing in diametral dimensionality, Siberskii Mathematicheskii Zhurnal II, 3 (1970), p. 512-525.

[5] E. D u b in s k y , Concrete sub spaces of nuclear Frêchet spaces (to appear).

[6] — and M. S. R a m a n u ja n , On X-nuclearity, Memoirs Amer. Math. Soc. 128 (1972) .

[7] M. S. R a m a n u ja n and T. T e r z io g lu , Diametral dimensions of Cartesian products, Stability of smooth sequence spaces and applications, J. Reine Angew.

Math. 280 (1976), p. 163-171.

[8] W. B. R o b in s o n , Belationships between X-nuclearity and pseudo-X-nuclearity, Trans. Amer. Math. Soc. 201 (1975), p. 291-303.

[9] T. T e r z io g lu , Die Diamétrale Dimension von lolcallconvexe Baumen, Collect.

Math. 20 (1969), p. 49-99.

[10] — Smooth sequence spaces and associated nuclearity, Proc. Amer. Math. Soc. 37 (1973) , p. 497-504.

[11] V. P. Z ah ar ju t a, On the isomorphism of cartesian products of locally convex spaces, Studia Math. 46 (1973), p. 201-221.

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