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Approximations of the Korovkin type in Banach lattices. (English) Zbl 1312.41032

Summary: Let \(E\), \(G\) denote two Banach lattices, and let \((T_n)\) be a sequence of continuous linear operators \(E \to G\). We prove that if \((T_n)\) satisfies the difference condition \(| T_n - T_m| x =| T_n x - T_m x|\) for all \(x \in E^+\), and if the sequence \((T_n x_0)\) converges for some \(x_0 \in E\), then \((T_n)\) converges pointwise on the principal ideal \(A_{x_0}\) generated by \(x_0\). This result allows us to strengthen essentially an approximate-spectral theorem of the Freudenthal type obtained recently by A. W. Wickstead.

MSC:

41A36 Approximation by positive operators
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B42 Banach lattices
47B65 Positive linear operators and order-bounded operators
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