Generalizing normality for operators on Banach spaces: Hyponormality. I (Q1072086)

In this paper the author attempts to define the notion of hyponormal element of a Banach algebra and to characterize hyponormal operators on a Hilbert space as an affirmative answer for the conjecture of S. Friedland. However the reviewer could not follow the ''decomposability'' of \(ua+v\) in proposition 1.6. (i.e., \(\| e^{ith}\| =1=\| e^{itk}\|\) for all \(t\in R\) implies \(\| e^{it(\alpha h+\beta k)}\| =1\) for all t,\(\alpha\),\(\beta\in R\), where h does not necessarily commute with k) and he could not understand why \(u\in H_ 2\) implies \(Pu=0\) in page 257 line 22 which seems to be the essential part of the proof for the conjecture of S. Friedland.