Numerically detectable hidden spectrum of certain integration operators

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 $\delta_1=1$.