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An extension of Knopp’s core theorem. (English) Zbl 0648.40004
Let $$A=(a_{nk})$$ be any matrix. Let $$B=(B_{jk})$$ be a normal matrix. The author gives necessary and sufficient conditions in order that whenever $$(\sum^{\infty}_{k=0}b_{jk}x_ k)_{j\in {\mathbb{N}}}$$ is bounded, $$(\sum^{\infty}_{k=0}a_{nk}x_ k)_{n\in {\mathbb{N}}}$$ exist for any bounded sequence $$(x_ n)$$ and be bounded and satisfy $$\limsup_{n}(\sum^{\infty}_{k=0}a_{nk}x_ k)\leq \limsup_{j}(\sum^{\infty}_{k=0}b_{jk}x_ k)$$. Theorem 2. Let B be a normal matrix. Then for a row finite matrix A, $$\limsup_{n}(\sum^{\infty}_{k=0}a_{nk}x_ k)\leq \limsup_{j}(\sum^{\infty}_{k=0}b_{jk}x_ k)$$ for all bounded sequences $$(x_ n)$$ iff $$AB^{-1}$$ is regular and almost positive. In a third theorem the author gives necessary and sufficient conditions in order that whenever $$(\sum^{\infty}_{k=0}b_{jk}x_ k)_{j\in {\mathbb{N}}}$$ is bounded, $$(\sum^{\infty}_{k=0}a_{nk}x_ k)_{n\in {\mathbb{N}}}$$ exist and satisfy $\limsup_{n}(\sum^{\infty}_{k=0}a_{nk}x_ k)\leq \inf_{t}\limsup_{j}(\sum^{\infty}_{k=0}b_{jk}x_ k+t),$ where B is a normal matrix, and the infimum is taken over all null sequences $$(x_ n)$$.
Reviewer: B.Crstici

##### MSC:
 40A25 Approximation to limiting values (summation of series, etc.) 40C05 Matrix methods for summability
##### Keywords:
Knopp’s core theorem; normal matrix
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##### References:
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