Download Narrow Operators on Function Spaces and Vector Lattices by Mikhail Popov PDF

By Mikhail Popov

Slender operators are these operators outlined on functionality areas that are "small" at symptoms, i.e. at {-1,0,1}-valued features. various works and examine papers exist on those, yet no coherent monograph but to put them in context. This publication supplies complete remedy of slim operators. It starts off with fundamentals after which systematically builds up the case. It additionally covers geometrical functions and Gaussian embeddings.

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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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