A version of Kalton’s theorem for the space of regular operators
Tóm tắt
In this note we extend some recent results in the space of regular operators [appeared in Bu and Wong (Indag Math 23:199–213, 2012), Bu et al. (Collect Math 62:131–137, 2011), and Li et al. (Taiwan J Math 16:207–215, 2012)]. Our main result is the following Banach lattice version of a classical result of Kalton: Let
$$E$$
be an atomic Banach lattice with an order continuous norm and
$$F$$
a Banach lattice. Then the following are equivalent: (i)
$$L^r(E,F)$$
contains no copy of
$$\ell _\infty $$
, (ii)
$$L^r(E,F)$$
contains no copy of
$$c_0$$
, (iii)
$$K^r(E,F)$$
contains no copy of
$$c_0$$
, (iv)
$$K^r(E,F)$$
is a (projection) band in
$$L^r(E,F)$$
, (v)
$$K^r(E,F)=L^r(E,F)$$
.
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