Numerically detectable hidden spectrum of certain integration operators

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 $r(T^{-1})\| T\| < 2$, $r( \cdot )$ being the spectral radius), and $\delta _{1}=1$ (no norm control of inverses). For more general integration operator $J_{\mu }f=\int _{[0,x\rangle }f d\mu$ 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 $\delta _{1}=1$.