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On the procedures ”min” and ”max” for operator ideals on Banach couples. (English) Zbl 0598.47048
Semin. Oper. Liniari Anal. Armonică, Univ. Timişoara 1, 9 p. (1984).
One extends the procedures from A. Pietsch, Operator ideals (1978; Zbl 0399.47039) of generating new ideals to the case of operator ideals on Banach couples.
The product of two operator ideals on Banach couples and the minimal kernel \({\mathcal A}^{\min}\) is defined. One shows the properties that result from the application of ”min” procedure to the particular ideals.
The quotient \({\mathcal A}^{-1}\circ {\mathcal B}\) is also defined and the ”max” procedure introduced. For closed operator ideals one shows that (\({\mathcal A}_ 1\wedge {\mathcal A}_ 2)^{\max}={\mathcal A}_ 1^{\max}\wedge {\mathcal A}_ 2^{\max}.\)
The following relations between two procedures are established: \[ ({\mathcal A}^{\min})^{\max}={\mathcal A}^{\max}\quad and\quad ({\mathcal A}^{\max})^{\min}={\mathcal A}^{\min}. \]
MSC:
47L10 Algebras of operators on Banach spaces and other topological linear spaces