On splitting polynomials with coefficients from commutative Banach algebras

The problem of decomposition of polynomials with coefficients from a unital commutative Banach algebra into a product of polynomials with coefficients in the same algebra is considered. Sufficient conditions for the existence of such decomposition and its construction are indicated. Some applications in the theory of Toeplitz operators on a circle and a torus are given. In particular, the equivalent regularization for a two-dimensional Toeplitz operator with a special symbol is derived. Equations generated by the corresponding Toeplitz operators in the spaces of measurable square-integrable vector-valued functions on the circle and measurable square summable functions are examined. This leads to the construction of equivalent regularizers for the described Toeplitz operators. The regularizer construction turns out to be equivalent to right canonical Wiener–Hopf factorization of some matrix function built from the symbol in a case of vector functions. An equivalent regularizer is built in explicit form for a two-dimensional Toeplitz operator with a special symbol.