Approximate identities, almost-periodic functions and Toeplitz operators (Q5945978)

Let \(\sigma_n (a)\) denote the Fejér-Cesàro means of the piecewise continuous function \(a(t)\) and \(T(\sigma_n (a))\) denote the Toeplitz operator and let also \({\mathbf{T}}=\{t\in {\mathbf{C}}: |t|=1\}\). The authors study the stability problem. The main result contains the following Theorem. Let \(a(t)=c_0+\sum_{j=1}^\infty( c_je^{\lambda_j\frac{t+1}{t-1}}+c_{-j}e^{-\lambda_j\frac{t+1}{t-1}})\), \(|t|=1\). Then the sequence of the Toeplitz operators of the form \(\{T(\sigma_n (a))\}_{n=1}^\infty\) is stable in \(H_2 (\mathbf{T})\) iff \(\inf_{u\in [0,\infty]} \inf_{t\in \mathbf{T}\backslash \{1\}} A(t,u)>0\) where \(A(t,u)=c_0+ \sum_{j=1}^\infty [( c_je^{\lambda_j \frac{t+1}{t-1}}+ c_{-j}e^{-\lambda_j \frac{t+1}{t-1}})t(1-\frac{\lambda_j}{u})_+]\).