1999/06/11 by C. Badea, M. Mbekhta
Mathematics · #math.FA #msc:47A10 #msc:47A20
published as Trans. Amer. Math. Soc. 351(1999), 2949-2960. · 15 pages, to appear in Trans. Amer. Math. Soc
arxiv created 1999/06/11 · arxiv updated 2009/11/30
Suppose that λ- T is left-invertible in L(H) for all λ∈ Ω, where Ω is an open subset of the complex plane. Then an operator-valued function L(λ) is a left resolvent of T in Ω if and only if T has an extension T, the resolvent of which is a dilation of L(λ) of a particular form. Generalized resolvents exist on every open set U, with U included in the regular domain of T. This implies a formula for the maximal radius of regularity of T in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by J. Zemánek is obtained.