Toeplitz operator

Summary

A Toeplitz operator on the Hardy space $H^2$ multiplies a function by a bounded symbol $\varphi$ on the circle and then projects back onto $H^2$. In the basis $1, z, z^2, \ldots$ its matrix is the infinite Toeplitz matrix of the Fourier coefficients of $\varphi$, constant along diagonals. Toeplitz operators link operator theory with complex analysis: their spectra, Fredholm indices and invertibility are read from the symbol.