Generalized Factorization of u-periodic Functions and Matrix Functions

In this chapter we confine ourselves to the setting of the space L _p (г₀), 1 <p <∞. First, we will give the needed definitions and facts from the theory of matrix Toeplitz operators. Then we will introduce the concept of the generalized factorization of a matrix valued function (matrix function for short) and formulate the corresponding invertibility theory for the operatorT(a)with matrix symbol a. Conceptually, the main section of this chapter is Section 5.3, devoted to the theory of generalized factorization of u-periodic matrix functions. After that we will consider new classes of Toeplitz operators with infinite index, first in the scalar and then in the matrix case.