PROBABILITY
AND
MATHEMATICAL STATISTICS
VoI. 2, Fa=. 2 (1982), p 157-160
ON BANACH SPACES
W. LINDE
( ~ N A ) ,V.
AND
M A N D R E K A R (EAST LANSING, MICHIGAN)
A. W E R O N (WROCLAW)
Abstract. For a real p (1 < p < 2) and its conjugate p' we
characterize Banach spaces E for which an operator T: L,, + E
is 8,-Radonifying iff T' is p-absolutely summing. In case p = 2 these
are exactly spaces of type 2 as was proved by Chobanjan and
Tarieladze [ I ] . For p < 2 the condition is much stronger because
these are spaces of stable type p isomorphic to a subspace of
some L,.
Let E be a real Banach space. For a real number p (I < p < 2) let
Lp be a separable Banach space of measurable functions having p-integrable
absolute value. Let l/p + lip' = 1. An operator T from Lpj into E is said
to be 0,-Radonifying if exp (- (1 T' a lip) is the characteristic function of
a Radon measure p on E. Here 6, is a cylindrical measure on L,, with
the characteristic function of the form exp (- llglIP), g E L , . Thus T is
8,-Radonifying iff T(0,) extends to a Radon measure on E . In this case
the Radon extension is a pstable symmetric measure on E. It turns out
that the set EP(L,,, E) of all 6,-Radonifying operators becomes a Banach
space under the equivalent norms
Let us recall that E is of stable type p if there exists a constant
c
> 0 such that for all x , , x,, ..., x , E~
158
W. Linde et-al.
<,
for each r with 0 < r < p, where
, {,, ..., c, is a sequence of i.i.d.
random variables with characteristic function exp (- (tlP).
If E and F are Banach spaces, then the operator T : E -r F is called
pabsolutely s~pnrpring (T€ITp(E,F)) if for some constant M and for each
xi, x,, ...,xn E E the inequality
holds. Denote by n p { T )the least such constait M.
The following relation between the 8,-Radonifying operators and the
plabsolutely summing operators is known in a more general version as
the celebrated L. Schwartz's duality theorem (cf. [2] and references therein):
PROPOSITION. If T E . Z ~ ( LE~) ,, then
,
T ' f 17p{E,Lp).
The converse implication does not hold in general. In the following
theorem we characterize Banach spaces for which it holds.
THEOREM.
Let 1 < p < 2 . Then the following two conditions on a Bunach
space E are equiua knt :
( 1 ) T E Zp(Lp,,E) if T' E 13,(E', Lp) for each space L,.
(2) E is of stabk type p and isomorphic to a subspace of some L,.
P r o of. (1) * (2). Let x,, x,, ..., X, E E. We define an operator T from
Lpvinto E by
Condition (1) implies the existence of a constant c > 0 such that
o,,(T) < cn,(T1). In addition, let us observe that the characteristic function
of the p-stable measure p defined by T is equal to the characteristic
n
xi ti.Namely,
function of the distribution of the E-valued random vector
i=l
n
P(Q) = exp(-IjT'a113
=
exp (-
C I<xi,a>Ip).
i=
1
Thus we have
which shows that E is of stable type p.
To prove that the space E is isomorphic to a subspace of some L,
we choose x,, x,, ..., x, and y,, y,, ...,y, belonging to E with the property
1 I(xi,a)jP.<
i= 1
i= 1
((yi,-a)lP for all ~ E E ' .
Radonifying operators
Now we define operators T and S from L,. into E by
1 TallP=
C
I(xi, a>lP and
/lS1allP=
I(yi, o)lP
for all a E E'.
i= 1
i= 1
The inequality 11 T' all < Ils'all for all a E E' implies zp( T ' ) <
each Banach sjiace is of stable cotype p for p < 2, we have
TL,
(S).Since
By Lindenstrauss-Pelcqnski's theorem ([4], Theorem 7.3) we claim that E
is isomorphic to a subspace of some L,.
(2)* (1). Consider an operator T: LPI+ E such that T' is pabsolutely
summing. Since E is isomorphic to a subspace of some L p , by Kwapieri's
theorem [33 we have T E It,(Lpj, E ) . I t follows from separability of the
space L, that there exists an isometric imbedding 3 from Lp into L,[O, 11.
Then TJ' is pabsolutely summing. By Kwapieh's theorem [2] there exists
a strongly measurable function q from LO, 11 into E with E flqIJP< co such
that
1
llT'aJIP= IIJT'aJIP=
j I{q (t),a)tPdt.
0
Since E is of stable type p, exp ( - jlT'allP) is (by [ 5 ] ) the characteristic
function of a Radon measure, i.e., T E Cp(L,, ,E ) , which completes the proof.
COROLLARY.
Let 1 < p < 2. Then the foIIowing two conditions on a Bunach space E are equivalent:
(1) T E ,Zp (L,, ,E) Sf and only if T' E U p(E', L,) for each space L,.
(2) E is of s t d l e type p and isomorphic to a subspace of some L,.
Remark. It is known by Rosenthal's theorem (see [ 7 ] ) that condition (2)
is equivalent to each of the following ones:
(3) E is isomorphic to a subspace of some L, and does not contain an
isomorphic copy of 1,.
(4) E is isombrphic to a subspace of some L, and. there exists a real r
(0 < r < p) such that the topologies of L, and L, coincide on E .
(5) There exists a real q (p < q < 2) such that E is isomorphic to a subspace of some L,.
Added in proof. After this note was completed, the authors were made
aware of the paper of D. H..Thang and N. D. Tien, On the extension
of stable cylindrical measures, Acta Math. Vietnam. 5 (1980), p. 169-177,
where an equivalent result was established. Methods of proofs are however
160
W. L i n d e et a].
different. We also refer the reader to our paper p-stable measures and
p-absolutet~~summing operators, p. 167-178 in: Lecture Notes in Math. 828
Springer Verlag, 1980, for some additional results on this subject.
REFERENCES
[I J S. A. C h o b anjan and V. I. Tariel adze, Gaussian characLerization of Banach &paces,
J. Multivariate Anal 7 (1977), p. 183-203
[2] S. Kwapleli, On a theorem of L Schwartz and tts application to absolutely summtny
operators, Studia Math. 3E (1970), p. 193-203
[3] - On operators factorrzable through L, space, Bull. Soc. Math. France 31-32(1972),
p. 215-225.
[4] J. L ~ n d e n s t r a u s sand A Pefczy riski, Absolutely summing operators m Ldspacrs and
their applications, Studia Math. 29 (1968), p. 275-326.
[5] D. M o u c h t a r ~ Sur
, E'exsr~en~e
d'une topologie du type Sazonov sur une rspace de Banach,
Serninaire Maurey-Schwartz 1975- 1976, Expose XVII.
[6J A. Pietsch, Operator ideals, VEB Deutscher Verlag der W~ssenschaften,Berlin 1978.
[7] H. F. Rosenthat, On subspaces o j L,, Ann. of Math. 97 (1973), p. 344-373.
W. Linde
Department of Mathematics
Friedrich Schillcr University
69 Jena, G.D.R.
V. Mandrekar
Department of Statistics and Probability
Michigan State University
East Lansing, MI 48824, U.S.A.
A. Weron
Institute of Mathematics
Wroclaw Technical University
pl. Grunwaldzki 13a
50-370 Wroclaw, Poland
Received on 7. 6. 1980