Divergence Almost Everywhere of a PointwiseComparison of Two Sequences of Linear Operators

✍ Scribed by S.V Konyagin

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 256 KB
• Volume: 90
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3072

✦ Synopsis

This paper deals with a negative result concerning a pointwise comparison of two quite general sequences of linear operators on the space of continuous functions on the torus or a segment 1997 Academic Press

Let E be the torus or a nongenerate segment of R. + denotes the Lebesque measure on E and & } & denotes the uniform norm on the space of continuous functions on E. Shapiro [6, p. 120] set up a problem of comparative pointwise behaviour of two sequences of linear operators on C(E). This question was studied in many works. For different pairs [A n ] and [B n ] of sequences of operators it was shown that the relationship

may fail almost everywhere (see [1 4]). Below we prove that this occurs for quite general sequences of operators.

Theorem. Let [A n ] and [B n ] be sequences of finite-dimensional linear bounded operators from the space C(E) to itself and let [I n ] be a nondecreasing sequence of positive numbers. Suppose also that A n ( f ) Ä f uniformly for any f belonging to a dense subset of C(E) and for some sequence of functions h n # C(E) we have lim inf n Ä +[x # E : A n (h n ; x)=B n (h n ; x)]=0.