Toeplitz matrices and determinants with Fisher-Hartwig symbols

AbstractThis paper is concerned with Toeplitz matrices generated by symbols of the form a(t) = Φr=1R |t−tr|2αr(−t)trβrb(t) (|t|=1) where t1,…, tR are pairwise distinct points on the unit circle, b is sufficiently smooth, b(t) ≠ 0 (¦t¦ = 1), and ind b = 0, (− t)trβr is defined as exp{iβrarg(− ttr)} with ¦arg(− ttr)¦ < π, and αr, βr are complex numbers satisfying ¦Re αr¦ < 12, ¦Re βr¦ < 12. Weighted Hardy spaces L+2(ϱ1) and L+2(ϱ2) are defined such that the Toeplitz operator T(a): L+2(ϱ1) → L+2 (ϱ2) is bounded and invertible and that the finite section method is applicable to T(a) considered as acting from L+2(ϱ1) onto L+2(ϱ2). These results are then applied to prove a conjecture of Fisher and Hartwig (Adv. in Chem. Phys.15 (1968), 333–353) which asserts that the determinants of the n × n sections of T(a) are asymptotically equal to G(b)n nqẼ(a) (n → ∞), where q = ∑ (αr2 − βr2) and G(b), Ẽ(a) are certain nonzero constants. Under quite general conditions on the smoothness of b, these constants are completely identified