Finite Rank Operators

This chapter is of a preliminary character. Here we accumulate results about the traces of finite rank operators as well as the determinants of operators of the form I + F, where F is an operator of finite rank Also, various formulas of trace and determinant are presented for operators of the form mentioned above. A variant of an axiomatic definition of the trace and determinant is proposed. This material is basic for the remaining chapters of the book where the determinant and trace are defined for more general classes of operators via limits of finite rank operators. The chapter also contains the trace and determinant of operator pencils and applications of the determinant to inversions of operators of the form I + F, where F has finite rank.