Stability of infinite ranges and kernels

Let z 7→ A(z) be a regular function defined on a connected set whose values are mutually commuting essentially Kato operators. Then the spaces R∞(A(z)) and N∞(A(z)) are constant. This generalizes results of Aupetit and Zemánek. Denote by B(X) the set of all bounded linear operators on a complex Banach space X. For T ∈ B(X) denote by N(T ) the null space and by R(T ) the range of T , respectively. Write also R∞(T ) = ⋂∞ k=1 R(T ) and N∞(T ) = ⋃∞ k=1 N(T ). It is well known that the spaces R∞(T − zI) and N∞(T − zI) remain constant for all z in a neighbourhood of zero for various classes of operators although the ranges R(T − zI) and kernels N(T − zI) do change, see [GK1], [H], [MO]. As it was observed by Aupetit and Zemánek [AZ], this phenomenon is closely related to the concept of regular functions. Denote by γ(T ) = inf{‖Tx‖ : dist {x,N(T )} = 1} the reduced minimum modulus of T . It is well known that γ(T ∗) = γ(T ), and γ(T ) > 0 if and only if T has closed range. Let G be a metric space, w ∈ G, and let A : G → B(X) be a continuous operatorvalued function. We say that A is regular at w if R(A(w)) is closed and A satisfies one of the following equivalent conditions: (i) the function z 7→ γ(A(z)) is continuous at w; (ii) lim infz→w γ(A(z)) > 0; (iii) the function z 7→ R(A(z)) is continuous at w in the gap topology; (iv) the function z 7→ N(A(z)) is continuous at w in the gap topology. Recall that the gap between two subspaces M, L ⊂ X is defined by δ̂(M, L) = max{δ(M, L), δ(L,M)} where δ(M, L) = sup x∈M ‖x‖≤1 dist {x, L}. Regular functions have been studied by a number of authors, see e.g., [Ma], [T], [J], [S], [M2]. By property (ii), the set of all regularity points is open. The regular functions are closely connected with the important class of Kato operators (sometimes also called semiregular operators). An operator T ∈ B(X) is called Kato if the function z 7→ T − z is regular at 0. It is well known, see e.g. [M2], p.113 that the following conditions are equivalent for an operator T with closed range: (i) T is Kato; (ii) N(T ) ⊂ R∞(T ); (iii) N∞(T ) ⊂ R(T ); (iv) N∞(T ) ⊂ R∞(T ); (v) N(T ) ⊂ z 6=0 N(T − zI); (vi) R(T ) ⊃ z 6=0 R(T − zI). * The second author would like to express his gratitude to the Technical University Berlin for warm hospitality during the preparation of this paper. The research was also supported by grant No. 201/03/0041 of GA ČR.