Geometry of the spectral semidistance in Banach algebras

Braatvedt, Gareth, and Brits, Rudi. "Geometry of the spectral semidistance in Banach algebras." Czechoslovak Mathematical Journal 64.3 (2014): 599-610. <http://eudml.org/doc/262137>.

@article{Braatvedt2014,
abstract = {Let $A$ be a unital Banach algebra over $\mathbb \{C\}$, and suppose that the nonzero spectral values of $a$ and $b\in A$ are discrete sets which cluster at $0\in \mathbb \{C\}$, if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that $a$ and $b$ are quasinilpotent equivalent if and only if all the Riesz projections, $p(\alpha ,a)$ and $p(\alpha ,b)$, correspond. For certain important classes of decomposable operators (compact, Riesz, etc.), the proposed formula reduces the involvement of the underlying Banach space $X$ in the computation of the spectral semidistance, and appears to be a useful alternative to Vasilescu’s geometric formula (which requires the knowledge of the local spectra of the operators at each $0\ne x\in X$). The apparent advantage gained through the use of a global spectral parameter in the formula aside, various methods of complex analysis can then be employed to deal with the spectral projections; we give examples illustrating the usefulness of the main results.},
author = {Braatvedt, Gareth, Brits, Rudi},
journal = {Czechoslovak Mathematical Journal},
keywords = {asymptotically intertwined; Riesz projection; spectral semidistance; quasinilpotent equivalent; asymptotically intertwined; Riesz projection; spectral semidistance; quasinilpotent equivalent},
language = {eng},
number = {3},
pages = {599-610},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Geometry of the spectral semidistance in Banach algebras},
url = {http://eudml.org/doc/262137},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Braatvedt, Gareth
AU - Brits, Rudi
TI - Geometry of the spectral semidistance in Banach algebras
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 599
EP - 610
AB - Let $A$ be a unital Banach algebra over $\mathbb {C}$, and suppose that the nonzero spectral values of $a$ and $b\in A$ are discrete sets which cluster at $0\in \mathbb {C}$, if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that $a$ and $b$ are quasinilpotent equivalent if and only if all the Riesz projections, $p(\alpha ,a)$ and $p(\alpha ,b)$, correspond. For certain important classes of decomposable operators (compact, Riesz, etc.), the proposed formula reduces the involvement of the underlying Banach space $X$ in the computation of the spectral semidistance, and appears to be a useful alternative to Vasilescu’s geometric formula (which requires the knowledge of the local spectra of the operators at each $0\ne x\in X$). The apparent advantage gained through the use of a global spectral parameter in the formula aside, various methods of complex analysis can then be employed to deal with the spectral projections; we give examples illustrating the usefulness of the main results.
LA - eng
KW - asymptotically intertwined; Riesz projection; spectral semidistance; quasinilpotent equivalent; asymptotically intertwined; Riesz projection; spectral semidistance; quasinilpotent equivalent
UR - http://eudml.org/doc/262137
ER -