On the kernel and cokernel of some toeplitz operators

Abstract

We show that the kernel and/or cokernel of a block Toeplitz operator T (G) are trivial if its matrix-valued symbol G satisfies the condition G(t−1)G(t)*=ING(t−1)G(t)*=IN. As a consequence, the Wiener–Hopf factorization of G (provided it exists) must be canonical. Our setting is that of weighted Hardy spaces on the unit circle. We extend our result to Toeplitz operators on weighted Hardy spaces on the real line, and also Toeplitz operators on weighted sequence spaces.