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Algebra i Analiz, 2016, Volume 28, Issue 6, Pages 70–83 (Mi aa1514)  

Research Papers

Numerically detectable hidden spectrum of certain integration operators

N. Nikolskiab

a St. Petersburg State University, Chebyshev Laboratory, 199178, St. Petersburg, Russia
b University of Bordeaux, France

Abstract: It is shown that the critical constant for effective inversions in operator algebras $alg(V)$ generated by the Volterra integration $Jf=\int_0^xf dt$ 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 $r(T^{-1})\|T\|<2$, $r(\cdot)$ being the spectral radius), and $\delta_1=1$ (no norm control of inverses). For more general integration operator $J_\mu f=\int_{[0,x>}f d\mu$ 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 $\delta_1=1$.

Keywords: effective inversion, visible spectrum, integration operator.

Funding Agency Grant Number
Russian Science Foundation 14-41-00010
This research is supported by the project “Spaces of analytic functions and singular integrals”, RSF grant 14-41-00010.


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English version:
St. Petersburg Mathematical Journal, 2017, 28:6, 773–782

Bibliographic databases:

Received: 25.06.2016
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Citation: N. Nikolski, “Numerically detectable hidden spectrum of certain integration operators”, Algebra i Analiz, 28:6 (2016), 70–83; St. Petersburg Math. J., 28:6 (2017), 773–782

Citation in format AMSBIB
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\pages 70--83
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\pages 773--782
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