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

Let E ⊕ F be a direct sum decomposition of a complex Banach lattice X. Garth Dales asked recently whether the equation ‖x+ y‖ = ‖ |x| ∨ |y| ‖ for all x ∈ E and y ∈ F 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 ‖x+ y‖ = ‖(|x|p + |y|p)1/p‖ for any p = 2. 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 S + iT with S and T hermitian always form a subalgebra of L(X), and that this subalgebra is (by the Vidav–Palmer theorem) isometrically a C∗-algebra. A particular conclusion is that, if E ⊕ F satisfies ‖x+ y‖ = ‖x+ eiθy‖ for all x ∈ E, all y ∈ F , and all θ ∈ [0, 2π), then it also satisfies the equation ‖x+ y‖ = ‖(|x|2 + |y|2)1/2‖ for all x ∈ E and y ∈ F .