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Numerically detectable hidden spectrum of certain integration operators


Author: N. Nikolski
Original publication: Algebra i Analiz, tom 28 (2016), nomer 6.
Journal: St. Petersburg Math. J. 28 (2017), 773-782
MSC (2010): Primary 47C05
DOI: https://doi.org/10.1090/spmj/1472
Published electronically: October 2, 2017
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Abstract: It is shown that the critical constant for effective inversions in operator algebras $ \mathrm {alg}(V)$ generated by the Volterra integration $ Jf=\int _{0}^{x}f\,dt$ in the spaces $ L^{1}(0,1)$ and $ L^{2}(0,1)$ are different: respectively, $ \delta _{1}=1/2$ (i.e., the effective inversion is possible only for polynomials $ T=p(J)$ with a small condition number $ r(T^{-1})\Vert T\Vert < 2$, $ r(\,\cdot \, )$ being the spectral radius), and $ \delta _{1}=1$ (no norm control of inverses). For more general integration operator $ J_{\mu }f=\int _{[0,x\rangle }f\,d\mu $ on the space $ L^{2}([0,1],\mu )$ with respect to an arbitrary finite measure $ \mu $, the following $ 0-1$ law holds: either $ \delta _{1}=0$ (and this happens if and only if $ \mu $ is a purely discrete measure whose set of point masses $ \mu (\{x\})$ is a finite union of geometrically decreasing sequences), or $ \delta _{1}=1$.


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Additional Information

N. Nikolski
Affiliation: Chebyshev Laboratory, St. Petersburg State University, 199178 St. Petersburg, Russia; University of Bordeaux, France
Email: nikolski@math.u-bordeaux.fr

DOI: https://doi.org/10.1090/spmj/1472
Keywords: Effective inversion, visible spectrum, integration operator
Received by editor(s): June 25, 2016
Published electronically: October 2, 2017
Additional Notes: This research is supported by the project “Spaces of analytic functions and singular integrals”, RSF grant 14-41-00010
Article copyright: © Copyright 2017 American Mathematical Society