6
ALEKSANDER PELCZYNSK 1
O.III. Absolutely summing operators and their relatives. Genera l reference s t o thi s
part ar e th e paper s b y Grothendiec k [Grl] , [Gr3] , Pietsc h [Pi] , Persson an d Pietsc h [P-P ]
and Seminair e Maurey-Schwart z 1972-1973, 1973-1974, 1974-1975 [M-S] .
B(X, Y) stand s fo r th e spac e of al l th e bounde d linea r operator s fro m a Banac h spac e
X int o a Banach spac e Y.
Let 1 p °° . Le t 5 be a compact Hausdorf f spac e an d le t fj. G [C(S)] J. B y /
we denote th e natura l injectio n o f C(S) int o L p(p)9 i.e. , the operato r whic h assign s to eac h
/ G C(S) it s n-equivalence class regarde d a s an element o f
L P(JJL).
I f E i s a closed linea r sub -
space o f C(S)
y
the n (unles s otherwise stated ) E denote s th e subspac e o f L
p(ji)
bein g th e
closure i n L
p(ji)-norm
o f i^piE) an d i^
p
denote s the restrictio n o f i^
p
t o E regarde d a s an
operator t o E .
DEFINITION 0.1. Le t 1 p °° . A bounded linea r operato r T: X Y i s p-integral
(resp. strictly p-integral) if ther e ar e a compact Hausdorf f spac e S, a fi G [C(S)] £ an d
bounded linea r operator s U: X C(S) an d V: Lp(p) * Y** (resp . V: Lp(ji)— Y) suc h
that
(0.4)
K
Y ' T = W MfPI/ (resp . T = K/ MtPt/)
where K
y
: 7 y* * denote s the canonica l embeddin g o f Y int o it s secon d dua l Y**.
A triple (U, V, i ) satisfying (0.4 ) is called a p-integral factorization o f T. Th e p-
integral norm o f 7 * is the quantit y
/p(r) = inf||U||II
V\\M\1/P
where th e infimu m i s extended ove r al l p-integral representation s o f T.
DEFINITION 0.2 . Le t 1 p °° . A bounded linea r operato r T: X Y i s p-absolutely
summing i f ther e ar e a compact Hausdorf f spac e S, a /z G [C(5) ] * , a closed linea r subspac e
E o f C(5) , and bounde d linea r operator s U: X E an d V: E^ Y suc h tha t
(0-5)
Vi
EpU=z
A triple (£/ , K , i^ ) satisfying (0.5 ) is called a p-absolutely summing factorization o f
r. Th e p-absolutely summing norm o f T is the quantit y
7rp(r) = in f HtfH
||F||||MII
1 /P
where th e infimu m i s extended ove r al l p-absolutely summin g factorization s o f T. W e shall
use the ter m "absolutel y summing " instea d o f "1-absolutely summing" .
The followin g resul t i s a slightly improve d versio n o f th e so-calle d Grothendieck-Pietsc h
theorem (cf . [Pi] , [P-P] , [Mt-P] , [P8]) .
THEOREM
0.4 . Let \p* and let T G B(X, Y).
(i) / / T is p-integral and j: X C(S0) is a fixed isometric isomorphic embedding of
X into C(5 0), S
0
arbitrary compact Hausdorff space, then there exists a fj. G [C(5 0)] % and
a linear operator V: L
p(p)
Y** such that
/ (r) = IN 1 / p, HK||=1, KT=Vi j .
Previous Page Next Page