In this paper, we define the generalized Kato spectrum of an operator, and obtain that the generalized Kato spectrum differs from the semi-regular spectrum on at most countably many points. We study the localized version of the single-valued extension property at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ 0 ∈ C in the case that λ 0 I − T admits a generalized Kato decomposition. From this characterization we shall deduce several results on cluster points of some distinguished parts of the spectrum.

Let X, Y be Banach spaces, A : X −→ Y and B, C : Y −→ X be bounded linear operators satisfying operator equation ABA = ACA. Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common properties of AC − I and BA − I in algebraic viewpoint and also obtained some topological analogues. In this note, we continue to investigate common properties of AC and BA from the viewpoint of spectral theory. In particular, we give an affirmative answer to one question posed by Corach et al. by proving that AC − I has closed range if and only if BA − I has closed range. 2010 Mathematics Subject Classification: Primary 47A05, 47A10.

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