Structure theorems for some classes of operators
I. Istrǎțescu · Proceedings of the Japan Academy Series A Mathematical Sciences · 1969
1. We consider bounded linear operators on a Hilbert space H. Denote by a(T), ap(T), at(T), ac(T) the spectrum, the point spectrum, the residual spectrum and the continuous spectrum respectively, byAn operator T is said to be hyponormal i. T* T--TT* >_ O, or equiva- lently if T*xll<_llTxll or every x e H.As in [1] an operator is said to be restriction-convexoid (reduction-convexoid) i the restriction of T to every invariant (invariant under T and T*) subspace is convexoid, where convexoid means that conv a(T)--cl W(T).In this Note we give some theorems on structure of hyponormal and restriction-convexoid operators whose spectrum lies on a convex curve.2. Our main result in this section is