Hermitian operators on complex Banach lattices and a problem of Garth Dales

ISSN： 0024-6107

Source： Journal of the London Mathematical Society, Vol.86, Iss.3, 2012-12, pp. : 641-656

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Abstract

Let EF be a direct sum decomposition of a complex Banach lattice X. Garth Dales asked recently whether the equation XyXy for all XE and yF implies that E and F are bands. We show that this is the case by using the theory of hermitian operators. We then show that the same result holds if we replace Dales's condition by Xy(X^py^p)^1/p for any p2. To do this, we develop a general theory of hermitian operators on a complex Banach lattice, showing in particular that the operators of the form SiT with S and T hermitian always form a subalgebra of (X), and that this subalgebra is (by the VidavPalmer theorem) isometrically a C-algebra. A particular conclusion is that, if E F satisfies XyXEⁱy for all XE, all yF, and all [0, 2), then it also satisfies the equation Xy(X²y²)^1/2 for all XE and yF.