Holomorphic functional calculus for operators on a locally convex space

One-variable holomorphic functional calculus is studied on the bornological algebra Lec(E) of all continuous linear oprators on a complete locally convex space E. It is proven that the following three basic notions of the theory are equivalent: (i) existence of projective resolvent of an operator T at a point λ₀, (ii) strict regularity of λ₀ for the operator T in the sense of [12, 13, 15], (iii) tamability of the operator (λ₀ ? T)^?1 (T if λ₀ = ∞), which means that there is a new equivalent system of seminorms on E, such that the operator is bounded in each of them.