Harmonic approximation of Toeplitz operators and index formulas
Let T(a) be the familiar Toeplitz operator on the Hardy space \(H^ 2_ m\) with matrix-valued symbol \(a\in L^{\infty}_{m\times m}\) and let \(h_ ra\) be the harmonic extension of a into the unit disk, \(0\leq r<1\). The paper under review is aimed at the investigation of the stable convergence of the sequence \(\{T(h_ ra)\}\) \((0\leq r<1)\) to T(a). In particular, there are proved the following results. 1. Let a be locally sectorial and let T(a) be invertible. Then \(\{T(h_ ra)\}\) converges stably to T(a). 2. Let a be locally sectorial. Then T(a) is Fredholm and ind T(a)\(=-wind \det (h_ ra)\) \((r\geq r_ 0).\) 3. Let a be locally sectorial and let the operators T(a) and T(ã) (ã(t):\(=a(1/t))\) be invertible. Then \[ \lim_{k\to \infty}D_ k(a)/D_{k-a}(a)=\lim_{r\to 1-0}\exp (\log \det h_ r\quad a)_ 0, \] where \(D_ k(a)\) denotes the Toeplitz determinant associated with k, and \((f)_ n\) denotes the n-th Fourier coefficient of f.
index formulas for Toeplitz operators via harmonic extension
scientific article; zbMATH DE number 3946961
scientific article