IGOR SIMONENKO AND THE MAGIC FASCINATION OF QUARTER-PLANE TOEPLITZ OPERATORS

Abstract This expository paper is written in remembrance of Igor Borisovich Simonenko (1935–2008) on the 90th anniversary of his birth. In the 1960s, Simonenko made fundamental contributions to one- and higher-dimensional convolution operators. The focus of this paper is on quarter-plane Toeplitz operators, the most prominent operators among multidimensional discrete convolutions. I will survey in a gentle way several selected topics around Simonenko’s Fredholm criterion of 1968, including the role played by local sectoriality and the invertibility problem for Toeplitz operators on the quarter-plane.