Generalised eigenvector expansion of infinite Toeplitz matrices with absolutely/completely monotone entries

We study the spectral theory of infinite Toeplitz matrices $T=(a_{k-l})$ under the assumption that $(a_k)$ and $(a_{-k})$ are completely monotone sequences. We derive expressions for generalised eigenvectors and prove a generalised eigenvector expansion of $T$. Even if the matrix $T$ is not normal, our expressions involve only eigenvalues and eigenvectors with real entries.