Aspects of (p.q) summing multipliers
Abstract
A sequence (uj )j∈N of operators in L(X, Y ) is a (p, q)-summing multiplier (or (p, q)-summing sequence of operators), in short (uj ) ∈ `πp,q (X, Y ), if there
exists a constant C > 0 such that, for any finite collection of vectors x1, x2, . . . xn in
X, it holds that
³Xn
j=1
kujxjk
p
´1/p
≤ C sup n³Xn
j=1
|x
∗
xj |
q
´1/q
; x
∗ ∈ BX∗
o
.
Some examples of these operators, inclusions between the spaces and connections
with spaces of multipliers are presented.∗
Mathematics Subject Classification (2000): 47B10.
Key words: Summing operators, vector-valued multipliers.
1. Introduction. Let X and Y be two real or complex Banach spaces and let
E(X) and F(Y ) be two Banach spaces whose elements are defined by sequences
of vectors in X and Y (containing any eventually null sequence in X or Y ). A
sequence of operators (un) ∈ L(X, Y ) is called a multiplier sequence from E(X)
to F(Y ) if there exists a constant C > 0 such that
°
°(ujxj )
n
j=1
°
°
F (Y )
≤ C
°
°(xj )
n
j=1
°
°
E(X)
for all finite families x1, . . . , xn in X.
The set of all of multiplier sequences is denoted by (E(X), F(Y )).
For the study of such multipliers for the cases of E(X) and F(Y ) corresponding to vector-valued Hardy spaces, vector-valued Bergman spaces, vector-valued
BMOA or spaces of vector valued Bloch functions the reader is referred to [AB1,
Bl1, Bl2, Bl3, Bl4].
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