zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] H. Alexander, J. Wermer: Several Complex Variables and Banach Algebras (3rd edition). Graduate Texts in Mathematics 35, Springer, New York, 1998. [2] R. M. Brits, H. Raubenheimer: Finite spectra and quasinilpotent equivalence in Banach algebras. Czech. Math. J. 62 (2012), 1101–1116. · Zbl 1274.46094 [3] I. Colojoară, C. Foiaş: Quasi-nilpotent equivalence of not necessarily commuting operators. J. Math. Mech. 15 (1966), 521–540. · Zbl 0138.07701 [4] C. Foiaş, F.-H. Vasilescu: On the spectral theory of commutators. J. Math. Anal. Appl. 31 (1970), 473–486. · Zbl 0175.13604 [5] K. B. Laursen, M. M. Neumann: An Introduction to Local Spectral Theory. London Mathematical Society Monographs. New Series 20, Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0957.47004 [6] B. Y. Levin: Lectures on Entire Functions. In collaboration with Y. Lyubarskii, M. Sodin, V. Tkachenko. Translated by V. Tkachenko from the Russian manuscript, Translations of Mathematical Monographs 150, American Mathematical Society, Providence, 1996. · Zbl 0856.30001 [7] M. Razpet: The quasinilpotent equivalence in Banach algebras. J. Math. Anal. Appl. 166 (1992), 378–385. · Zbl 0802.46064 [8] F.-H. Vasilescu: Analytic Functional Calculus and Spectral Decompositionsk. Mathematics and Its Applications (East European Series) 1, D. Reidel Publishing, Dordrecht, 1982, translated from the Romanian. [9] F.-H. Vasilescu: Some properties of the commutator of two operators. J. Math. Anal. Appl. 23 (1968), 440–446. · Zbl 0159.43402 [10] F. H. Vasilescu: Spectral distance of two operators. Rev. Roum. Math. Pures Appl. 12 (1967), 733–736. · Zbl 0156.38204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.