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By defining and applying the restricted topology, we have investigated certain connections between the boundary spectrum, the exponential spectrum, the topological boundary of the spectrum and the connected hull of the spectrum (see Mouton and Harte [‘Linking the boundary and exponential spectra via the restricted topology’, J. Math. Anal. Appl.454 (2017), 730–745]). We now solve a remaining problem regarding the restricted connected hull.
Harte (1982, Math. Z.179, 431–436) initiated the study of Fredholm theory relative to a unital homomorphism T: A → B between unital Banach algebras A and B based on the following notions: an element a ∈ A is called Fredholm if 0 is not in the spectrum of Ta, while a is Weyl (Browder) if there exist (commuting) elements b and c in A with a = b + c such that 0 is not in the spectrum of b and c is in the null space of T. We introduce and investigate the concepts of r-Fredholm, r-Weyl and r-Browder elements, where 0 in these definitions is replaced by the spectral radii of a and b, respectively.
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