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Loïc Foissy 《Advances in Mathematics》2011,226(6):4702
We consider systems of combinatorial Dyson–Schwinger equations in the Connes–Kreimer Hopf algebra HI of rooted trees decorated by a set I. Let H(S) be the subalgebra of HI generated by the homogeneous components of the unique solution of this system. If it is a Hopf subalgebra, we describe it as the dual of the enveloping algebra of a Lie algebra g(S) of one of the following types:
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g(S) is an associative algebra of paths associated to a certain oriented graph. 2.
Or g(S) is an iterated extension of the Faà di Bruno Lie algebra. 3.
Or g(S) is an iterated extension of an infinite-dimensional abelian Lie algebra.
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Zhang Xiao-Dong 《Indagationes Mathematicae》1996,7(4):559
Extension properties of compact positive operators on Banach lattices are investigated. The following results are obtained:
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(1) Any compact positive operator (any compact lattice homomorphism, resp.) from a majorizing sublattice G of a Banach lattice E into another Banach lattice F can be extended to a compact positive operator (a compact lattice homomorphism, resp.) from E into F; 2.
(2) Any compact positive operator defined on a closed majorizing sublattice G of a Banach lattice E has a compact positive extension on E that preserves the spectrum (a necessary modification is needed).