This paper gives some very elementary proofs of results of Aupetit, Ransford and others on the variation of the spectral radius of a holomorphic family of elements in a Banach algebra. There is also some brief discussion of a notorious unsolved problem in automatic continuity theory.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc75-0-1,
author = {Graham R. Allan},
title = {Some simple proofs in holomorphic spectral theory},
journal = {Banach Center Publications},
volume = {75},
year = {2007},
pages = {9-15},
zbl = {1121.46040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc75-0-1}
}
Graham R. Allan. Some simple proofs in holomorphic spectral theory. Banach Center Publications, Tome 75 (2007) pp. 9-15. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc75-0-1/