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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Solvability of systems of linear operator equations

Authors: Rong Qing Jia, Sherman Riemenschneider and Zuowei Shen
Journal: Proc. Amer. Math. Soc. 120 (1994), 815-824
MSC: Primary 47A50; Secondary 39A70
MathSciNet review: 1169033
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Abstract: Let $ G$ be a semigroup of commuting linear operators on a linear space $ S$ with the group operation of composition. The solvability of the system of equations $ {l_i}f = {\phi _i},\;i = 1,\, \ldots \,,\,r$, where $ {l_i} \in G$ and $ {\phi _i} \in S$, was considered by Dahmen and Micchelli in their studies of the dimension of the kernel space of certain linear operators. The compatibility conditions $ {l_j}{\phi _i} = {l_i}{\phi _j},i \ne j$, are necessary for the system to have a solution in $ S$. However, in general, they do not provide sufficient conditions. We discuss what kinds of conditions on operators will make the compatibility sufficient for such systems to be solvable in $ S$.

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Keywords: Systems of operator equations, multivariate approximation, polynomial ideals, linear partial differential equations, linear partial difference equations
Article copyright: © Copyright 1994 American Mathematical Society

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