Holomorphic functional calculus for operators on a locally convex space

Schauder decompositions and functional calculus

“…The points outside the ultraspectrum are called strictly regular points, and the set of such points is denoted by ρ u (T ). In general we have that σ u (T ) ⊆ σ r (T ), when σ r (T ) is taken with respect to the bounded structure, on the operator algebra, formed by the equicontinuous sets [2,Corollary 13]. In fact, from Remark 11 in [2] we have that a point λ ∈ C is strictly regular for an operator T ∈ L(H) if and only if T − λ is invertible and for any r > 0 there exist M, R > 0 such that (T − λ) −n f r M n f R for all n 1 and f ∈ H. (Thus in order that λ ∈ ρ r (T ), we must be able to find an M that works for all r.) Next, λ = ∞ is strictly regular if and only if for any r > 0 there exist M, R > 0 such that T n f r M n f R for all n 1 and f ∈ H. (Again, in order that ∞ ∈ ρ r (T ), there must be an M that works for all r.) From this we obtain:…”

“…If we multiply this identity from the right by p α , and use the fact that p i α i p α = p α , we see that we must have that (a 1 α 1 − λ 1 )S 1 + · · · + (a n α n − λ n )S n is the identity on P α . Now, since any regular element is tamable [2], (S 1 , . .…”

“…Another type of spectrum, the ultraspectrum σ u (T ) ⊆Ĉ, for an operator T , is introduced in [2]. The points outside the ultraspectrum are called strictly regular points, and the set of such points is denoted by ρ u (T ).…”

“…However, functional calculus for (i) unbounded Banach space operators, and for (ii) continuous operators on nonnormable spaces have also been investigated (see e.g. [2,3,13]), the latter however quite rarely. The problem in these lines of investigations usually relates to the spectrum.…”

The Growth Bound for Strongly Continuous Semigroups on Fréchet Spaces

“…For A ∈ L(X) the above definition of resolvent and spectrum coincides with the those given by Allan in .7)], they studied the condition of Lemma 6 as an additional property of points in the resolvent. Arikan, Runov, Zahariuta [3] introduced the ultraspectrum for operators in A ∈ L b (X). In their notation, λ ∈ C is a strictly regular point of A if R(λ, A) exists in L(X) and is tamable [3, p. 29], i.e., there exists a fundamental system of seminorms Γ for X such that for all q ∈ Γ there exists C 0 such that q(R(λ, A)x) Cq(x) holds for all x ∈ X.…”