Modified Finite Sections for Toeplitz Operators and Their Singular Values

The topic of this paper is the study of modified finite sections of Toeplitz operators and their singular values. We prove the splitting property for the singular values and consider two important consequences. We show that the kernel dimension of a Fredholm Toeplitz operator with a piecewise continuous matrix-valued generating function can be extracted from the singular values behavior of the modified sections. Second, we generalize the results on asymptotic Moore--Penrose invertibility of Heinig and Hellinger [Integral Equations Operator Theory, 19 (1994), pp. 419--446] to piecewise continuous generating functions.