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Compact -friendly operators

Purdue University
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  • Mathematics
  • Mathematics


Chapter 2 deals with compact-friendly operators in any Banach lattice. In first two sections we extend the celebrated Lomonosov theorem [Lom73] to real Banach spaces under some additional assumptions on the spectrum of an operator. Then we apply the obtained version to positive operators and obtain several theorems about invariant subspaces of compact-friendly operators. In one of them we show that if T is a positive operator that commutes with a positive operator S which commutes with a positive operator R majorized by a compact operator, then T has a closed non-trivial invariant subspace. In the remaining sections we investigate the set of compact-friendly operators. We prove, in particular, that it is majorizing and in many cases is order dense. This answers a question posed by Y. Abramovich, C. Aliprantis, and O. Burkinshaw in [AAB97]. ^ Chapter 3 is devoted to investigation of the relation between the well known classes of multiplication operators and weighted composition operators on the one hand, and our new class of compact-friendly operators on the other. We consider these classes of operators in function spaces and spaces of continuous functions. While the case of weighted composition operators is covered only partially, the case of multiplication operators is settled completely. We obtain a complete characterization of multiplication compact-friendly operators in function spaces, answering a question in [AABW99]. An abstract version of this result is obtained as well. Specifically, we prove that a positive central operator M is compact-friendly if and only if it has a positive eigen-vector. ^

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