Algebraic theory of atkinson operators

Notations: In this paper E denotes a linear space, δ(E) the set of all endomorphisms  of E , L(E) the set of all continuous endomorphisms of E if E is equipped with a vector  space topology.  If  T ∈δ (E),  then                        

N(T) ={x:    Tx= 0}

B(T) = {Tx:  x   ∈ E }

∝ (T) =  dim N (T)

β (T) = codim B(T)

Based on the wo rk of Fredholm,  F. Riesz and Noether, Atkinson defined the abstract  concept of a Fredholm operador on a Banach space E : T ∈ ?( E) is called a Fredholm  operator if ∝ (T) and β(T)are both finite and B(T) is closed. As was shown later by Kato, β (T)  <∞ already implies that B(T) is close. When considering continuous operators T on a Banach space E for wich only one of the numbers  ∝ (T) β (T) was supposed to be finite, Atkinson remarked that the hypothesis B(T) is closedil was not sufficient to establish a satisfying theory. He therefore introduced the stronger hypothesis of relative regularity: T ∈ ?( E)  is calles relatively regular if there exists a S ∈ ?( E)  such that

T S T = T