ISSN:
1572-9281
Keywords:
Banach lattice
;
compact-friendly operators
;
invariant subspace
;
multiplication operators
;
positive operator
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The main results obtained are: – A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T ∈ Lr(X,Y): ||T||r ≤ 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y). – A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y). – A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm. – A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm. Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance: – If X = Lp(μ) and Y = Lq(ν), where 1 〈 p,q 〈 ∞, then Lr(X,Y) is a first category subset of L(X,Y).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1009743831883
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