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Be a finite atomless measure space, 0 < p < 1 and X be an F-space. Lp . /; X/ is a narrow operator then T D 0. Lp . /; X/ is an AM-compact operator then T D 0. 2 The separable quotient space problem In the first edition of his book [122] (1972) Rolewicz asked whether there exists an infinite dimensional F-space with no separable infinite dimensional quotient space. This question was answered in [112] using techniques of narrow operators. ˛ with ˛ > 0 has no separable quotient space. 9 ([110, 112]).

10. 10 is essential, since L1 . / is an example of a Köthe F-space on . ; †; / in which the set of all simple functions is dense, however the norm is not absolutely continuous. 12. Let E be a Köthe F-space with an absolutely continuous norm on the unit on a finite atomless measure space . ; †; / and X an F-space. E; X/ is narrow. F Proof. 14, we decompose D ˛2M ˛ so that for /; j†. ˛ / every ˛ 2 M, the measure spaces . ˛ ; †. ˛P are isomorphic for some "˛ > 0 such that ˛2M "˛ D . /, where M is the Maharam set of the measure space .

X; Y /. Proof. We set W Y ! y1 C y2 / D y1 , where y1 2 Y1 , y2 2 Y2 . X; Y / the operator S W X2 ! T x/, for each x 2 X2 is linear and continuous. Hence, by the assumption, S D 0, that is T x 2 Y2 for each x 2 X2 . 12 has the following two immediate consequences. 13. Y2 ; X1 / D ¹0º. X; Y /. 14. X; Y / D ¹0º, and let Z D X ˚ Y . Then TX D X for every automorphism T W Z ! Z. The following statement easily follows from the definition of a narrow operator. 15. Let X be a Köthe F-space on a finite atomless measure space .

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