The (n,ϵ)-pseudospectrum of an element of a Banach algebra

Let A be a complex unital Banach algebra, a∈A, n∈Z+ and ϵ>0. The (n,ϵ)-pseudospectrum Λn,ϵ(a) of a is defined asΛn,ϵ(a):=σ(a)∪{λ∉σ(a):‖(λ−a)−2n‖1/2n≥1ϵ}. Here σ(a) denotes the spectrum of a. The usual pseudospectrum Λϵ(a) of a is a special case of this, namely Λ0,ϵ(a). It is proved that (n,ϵ)-pseudospectrum approximates the closed ϵ-neighbourhood of spectrum for large n. Further, it has been shown that (n,ϵ)-pseudospectrum has no isolated points, has a finite number of connected components and each component contains an element from σ(a). Some examples are given to illustrate these results.