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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
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