ON A CLASS OF GENERALIZED FREDHOLM OPERATORS, I
Let X be an infinite-dimensional complex Banach space and let C(X) denote the Banach algebra of all bounded linear operators on X. We write $ g (X) for the following class of operators: TST = T and I -ST -TS is Fredholm}. Each Fredholm operator belongs to <$g(X). Operators in d> g (X) we call generalized Fredholm operators. In this paper we investigate the class <P g (X). Terminology and introduction Throughout this paper X denotes an infinite-dimensional complex Banach space and A denotes a
... and A denotes a complex algebra with identity e 0 (the considerations in A will be purely algebraic). C(X) denotes the set of all bounded linear operators on X. For T G £(X) write N(T) and T(X) for the kernel and the range of T, respectively. T 6 C(X) is called Fredholm if dim N(T) and codim T(X) are both finite. We write $(X) for the set of all Fredholm operators on X. If T € $(X) then the index ind(T) of T is defined as An element t € A is called relatively regular if tst = t for some s € A. In this case we call s a pseudo-inverse of t. It is well known that Fredholm operators are relatively regular , § 74. Furthermore we have for T € £(X): T is relatively regular if and only if N(T) and T(X) are continuously projectable. 1991 Mathematics Subject Classification: 47A11.