Perturbation and Coperturbation Functions Characterising Semi-fredholm-type Operators

Certain norm-related functions have been considered in order to obtain characterisations and perturbation results for various classes of semi-Fredholm-type operators. In the §2, by means of what we have called a perturbation function, we present a general approach to the question of obtaining characterisations and perturbation theorems for F + and strictly singular operators. To this end, for each perturbation function ψ, we construct functions Γψ and ∆ψ, which are similar to Γ and ∆ introduced by Cross [1]. In §3, we study the F − operator and strictly cosingular operator, in a similar way to what we have done before for F + and strictly singular operators, but using the notion of a coperturbation function. For this purpose, we define, for each coperturbation function , the functions (Γ )« and (∆ )« which are similar to Γ« and ∆« considered by Labuschagne [7] (see also [10]). Let X and Y be infinite dimensional normed spaces. The collections of infinite dimensional, closed infinite codimensional, finite dimensional, finite codimensional and closed finite codimensional subspaces of X are respectively denoted by I(X ), Ic(X ), F(X ), C(X ) and P(X ). L(X, Y ) denotes the class of linear operators defined on a subspace D(T ) of X having range in Y. The range and null space of T are respectively denoted by R(T ) and N(T ). For a subspace M of X, iM denotes the element of L(X, X ) which is the canonical injection of M into X, with the quotient map from X onto X}M being denoted by qM. We write JX for the injection of X into its completion X C and X« is the dual of X. Let J denote the operator in L(D(T ), X ) that is the identity injection of D(T ) into X. Then the conjugate T « of T is defined to be the conjugate of TJ in the sense of [5]. Square brackets will be used to indicate that only everywhere-defined operators are considered, for example, PK[X, Y ] denotes the class of everywhere-defined precompact operators in L(X, Y ). Continuous everywhere-defined operators will be referred to as bounded operators.