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Asymptotics for Toeplitz operators with piecewise quasicontinuous symbols and related questions. (English) Zbl 0609.47037

Let T be the unit circle in the complex plane \({\mathbb{C}}\) and B a closed subalgebra of \(L^{\infty}_{N\times N}=(L^{\infty}(T))_{N\times N}\), the Banach algebra of \(N\times N\) matrices with entries form \(L^{\infty}(T)\). for \(a=(a_{ij})^ N_{i,j=1}\in L^{\infty}_{N\times N}\) the block Toeplitz operator \(T(a):\ell^ 2_ N\to \ell^ 2_ N\) is defined by \(T(a)=(T(a_{ij}))^ N_{i,j=1}\) with each entry a Toeplitz operator on \(\ell^ 2\). Let \(p_ n:\ell^ 2\to \ell^ 2\) be the projection onto the first \(n+1\) coodinates and \(P_ n\) the \(N\times N\) matrix with \(p_ n\) along the diagonal. We say that the finite section method is applicable to T(a) is the operators \(T_ n(a)=P_ nT(a)P_ n|_{ran P_ n}\) are invertible for large enough n (say \(n\geq n_ 0)\) and if \(\sup_{n\geq n_ 0}\| (P_ nT(a)P_ n)^{-1}P_ n\| <\infty\). Denote the determinant of the finite Toeplitz matrix \(T_ n(a)\) by \(D_ n(a)\). Also the harmonic extension of \(a\in B\) on the circle \(\{| z| =r\}\) is denoted by \(a_ r.\)
The author uses Banach algebra techniques to prove the first Szegö limit theorem for Toeplitz determinants generated by piecewise quasicontinuous matrix-valued functions. It is as follows.
Let \(a\in B=(PQC)_{N\times N}\) and the operators \(T(a)\), \(T(\tilde a)\) be invertible. Then \[ \lim_{n\to \infty}D_ n(a)/D_{n- 1}(a)=\lim_{r\to 1^-}\exp (\log \det a_ r)_ 0. \] It is also shown that the finite section method is applicable to a Toeplitz operator T(a) with \(a\in (PQC)_{N\times N}\) if and only if \(T(a)\) and \(T(\tilde a)\) are invertible \([\tilde a(t)=a(1/t)]\). The Fredholm theory of Toeplitz operators is also discussed and an index formula for Fredholm block Toeplitz operators with piecewise quasicontinuous symbols is derived.
Reviewer: K.Seddighi

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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