Geodesic
Geometry of the spectral semidistance in Banach algebras
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 599-610
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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 $0\not =x\in X$). 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.
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 $0\not =x\in X$). 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.
DOI : 10.1007/s10587-014-0121-x
Classification : 46H05, 47A05, 47A10
Keywords: asymptotically intertwined; Riesz projection; spectral semidistance; quasinilpotent equivalent 
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Braatvedt, Gareth; Brits, Rudi. Geometry of the spectral semidistance in Banach algebras. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 599-610. doi: 10.1007/s10587-014-0121-x

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