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Geometry of the spectral semidistance in Banach algebras. (English) Zbl 1349.46047
For a complex Banach algebra \(A\) with identity \(\mathbf 1\) and \(a,b\in A\) let \(L_a, R_b\), and \(C_{a,b}\) denote the operators acting on \(A\) by the formulas \[ L_ax=ax, \quad R_bx=xb,\quad C_{a,b}x=(L_a-R_b)x \quad \text{for} \;x\in A. \] Let also \(\varrho : \,A\times A\rightarrow {\mathbb R}\) be defined as \[ \varrho(a,b)=\limsup_{n}\| C_{a,b}^n \mathbf 1\|^{1/n} \] and let \[ \rho(a,b)=\sup\{\varrho(a,b),\varrho(b,a)\}. \] The quantity \(\rho(a,b)\) is called the spectral semidistance. If \(\rho(a,b)=0\), then \(a\) and \(b\) are quasinilpotent equivalent. The authors obtain some formulas for the spectral semidistance in cases when the spectra of \(a\) and \(b\) are finite or infinite discrete sets such that nonzero spectral points can cluster only at zero. The results of the paper are related to Vasilescu’s geometric formula for the spectral semidistance of decomposable Banach space operators [F. H. Vasilescu, Rev. Roum. Math. Pures Appl. 12, 733–736 (1967; Zbl 0156.38204)].
The main result of the paper under review reads as follows:
Theorem. Let \(\sigma'(a)=\{\lambda_1,\lambda_2,\ldots\}\) and \(\sigma'(b)=\{\beta_1,\beta_2,\ldots\}\) be the sets of the nonzero spectral points of \(a\) and \(b\) and let \(\{p_1,p_2,\ldots\}\) and \(\{q_1,q_2,\ldots\}\) be the corresponding Riesz projections. Suppose that \(\sigma'(a)\) and \(\sigma'(b)\) cluster at zero if anywhere. Then \(\varrho(a,b)\) takes at least one of the following values:
(1) \(\varrho(a,b)=\sup\{|\lambda_i-\beta_j| : \, p_iq_j\neq0\}\), or
(2) \(\varrho(a,b)=|\lambda_i|\) for some \(i\in{\mathbb N}\), or
(3) \(\varrho(a,b)=| \beta_i|\) for some \(i\in{\mathbb N}\).
Moreover, \(\varrho(a,b)=0\) if and only if the spectra and the corresponding Riesz projections of \(a\) and \(b\) coincide.
MSC:
46H05 General theory of topological algebras
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A10 Spectrum, resolvent
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References:
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