The spectra of preconditioned Toeplitz matrix sequences can have gaps

Different than for the case of Toeplitz matrix sequences $\{T_n(f)\}$, $f\in L^1$, we can prove that the closure of the union of all the spectra of preconditioned matrix sequences of the form $\{T_n^{-1}(g)T_n(f)\}$, $f,g\in L^1$, $g\ge 0$, can have gaps if the essential range of f/g is not connected. The result has important consequences on the practical use of band Toeplitz preconditioners widely used in the literature both for (multilevel) ill-conditioned positive definite and (multilevel) indefinite Toeplitz linear systems