Interpolation of compact operators on Banach triples. (English) Zbl 1004.46048

Ceauşu, Traian; Cofan, Nicolae; Stan, Ilie

An. Univ. Timiş., Ser. Mat.-Inform. 36, No. 2, 205-218 (1998).

Summary: Let \(\overline{A}=(A_0,A_1,A_2)\) and \(\overline{B}=(B_0,B_1,B_2)\) be \(LP\)-triples (Lions-Peetre triples) of Banach spaces and let \(T\) be a linear operator. If \(T\colon A_0\to B_0\) is compact and \(T\colon A_i\to B_i\) is bounded (\(i=1\), \(2\)), then also \(T\colon\overline{A}_{f,p}\to \overline{B}_{f,p}\) is compact, where \(f\) belongs to the class \(B_{K}^{2}\) and \(1\leq p\leq \infty\).

MSC:

46M35	Abstract interpolation of topological vector spaces
47B07	Linear operators defined by compactness properties
46B70	Interpolation between normed linear spaces

Keywords:

Banach triples; interpolation of Banach space operators; Lions-Peetre triples