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Analytic perturbation of the Taylor spectrum


Author: Zbigniew Slodkowski
Journal: Trans. Amer. Math. Soc. 297 (1986), 319-336
MSC: Primary 47A56; Secondary 32A99, 47A10, 47D99
DOI: https://doi.org/10.1090/S0002-9947-1986-0849482-9
MathSciNet review: 849482
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Abstract: Let $ {T_1}(z), \ldots ,{T_m}(z)$, $ z \in G \subset {{\mathbf{C}}^k}$, be analytic families of bounded operators in a complex Banach space $ X$, such that for each $ z \in G$ the operators $ {T_i}(z)$ and $ {T_j}(z)$, $ i,j = 1, \ldots ,n$, commute. Main result: If $ K(z)$ denotes the Taylor spectrum of the tuple $ ({T_1}(z), \ldots ,{T_m}(z))$, then the set-valued function $ K:G \to {2^{{\mathbf{C}}m}}$ is analytic. Analyticity of such set-valued functions is defined here by a simultaneous local maximum property of $ k$-tuples of complex polynomials on the graph of $ K$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0849482-9
Keywords: Analytic multifunction, local maximum property, joint spectrum, Taylor spectrum, exact complex, analytic perturbation
Article copyright: © Copyright 1986 American Mathematical Society

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