11,815 research outputs found
Free integro-differential algebras and Groebner-Shirshov bases
The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions.
In this paper we apply the method of Groebner-Shirshov bases to construct the free (noncommutative) integro-differential algebra on a set. The construction is from the free Rota-Baxter algebra on the free differential algebra on the set modulo the differential Rota-Baxter ideal generated by the noncommutative integration by parts formula. In order to obtain a canonical basis for this quotient, we first reduce to the case when the set is finite. Then in order to obtain the monomial order needed for the Composition-Diamond Lemma, we consider the free Rota-Baxter algebra on the truncated free differential algebra. A Composition-Diamond Lemma is proved in this context, and a Groebner-Shirshov basis is found for the corresponding differential Rota-Baxter ideal
Order continuous extensions of positive compact operators on Banach lattices
Let and be Banach lattices. Let be a vector sublattice of and
be an order continuous positive compact (resp. weakly
compact) operators. We show that if is an ideal or an order dense
sublattice of , then has a norm preserving compact (resp. weakly
compact) positive extension to which is likewise order continuous on .
In particular, we prove that every compact positive orthomorphism on an order
dense sublattice of extends uniquely to a compact positive orthomorphism on
.Comment: 7 page
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