ON A PROBLEM OF N.SALINAS

This paper presents a discussion on a problem suggested by N.Salinas(J.Operator Theory,Vol.10,1983),i.e.,if T=(T_1,T_2,…,T_n)is a commuting n-tuple ofhyponormal operators on a complex Hilbert space,do we have(1)(2)δ(T-μ)=dist(μ,σ_l(T)),μ∈C~n?It is proved that the problems(1)and(2)are both true for some commuting n-tuples ofsemihyponormal operators.In general case,problem(1)has a negative answer.As an ap-plication,we point out that there exists a commuting n-tuple of hyponormal operatorsT(n≥2),such that|λ|<,where σ_T(T)denotes J.L.Taylor'sjoint spectra of T,even if the condition of Taylor's joint spectra of T is given.

ON A PROBLEM OF N.SALINAS