Spectral continuity using ν-convergence

Let T,TnT,Tn, n∈Nn∈N, be bounded linear operators defined on a Banach space X   such that {Tn}{Tn} converges to T in ν  -convergence (this new mode of convergence was observed by Mario Ahues). In this paper, sufficient conditions are given for the convergence γ(Tn)→γ(T)γ(Tn)→γ(T), where γ∈{σ,σap}γ∈{σ,σap}. Also we give some conditions for a bounded operator S   in order to have the stability of convergence: γ(Tn+S)→γ(T+S)γ(Tn+S)→γ(T+S). Among other things, we show that if 0 is an accumulation point of σ(T)σ(T) then σ(Tn)→σ(T)σ(Tn)→σ(T) when Tn,TTn,T satisfy the growth condition (G1)(G1).