Given ⊂ ℕ, let ̂ be the set of all positive integers m for which is hermitian whenever h is an element of a complex unital Banach algebra A with hⁿ hermitian for each n ∈ . We attempt to characterize when (i) ̂ = ℕ, or (ii) ̂ = . A key tool is a Müntz-type theorem which gives remarkable conclusions when 1 ∈ and ∑ 1/n: n ∈ diverges. The set ̂ is determined by a single extremal Banach algebra Ea(). We describe this extremal algebra for various .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-1-6,
author = {M. J. Crabb and J. Duncan and C. M. McGregor and T. J. Ransford},
title = {Hermitian powers: A M\"untz theorem and extremal algebras},
journal = {Studia Mathematica},
volume = {147},
year = {2001},
pages = {83-97},
zbl = {0999.46021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-1-6}
}
M. J. Crabb; J. Duncan; C. M. McGregor; T. J. Ransford. Hermitian powers: A Müntz theorem and extremal algebras. Studia Mathematica, Tome 147 (2001) pp. 83-97. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-1-6/