SOME VARIANTS OF THE KLEINECKE-SHIROKOV AND WEBER’S THEOREMS

Abstract. For bounded linear operators T: X 7 → Y and S: Y 7→ Z on Banach spaces the condition kerT ∩ R(T) = 0 is equivalent to the equality ker(ST) = kerT; when X = Y = Z and T = Sn, this is the familiar condition that the operator S has ascent ≤ n. Stronger conditions would replace the range R(T) of T by its closure, either in the norm or in some weaker topology; weaker conditions would ask that the intersection of kerS∩R(T) with some subspace of Y was in some sense nearly zero. Thus the celebrate Kleinecke-Shirokov theorem [1, 6] states that if X = Y = Z = A for a Banach algebra A and S = T = δa: x 7 → ax − xa is an inner derivation on A, then ker(S) ∩ R(T) ⊆ Q, where Q = QN(A) is the quasinilpotents in A. Weber [8] showed for same S and T then when A = B(H) for separable Hilbert space H then ker(S) ∩ R(T)w ∩ J ⊆ Q, J = K(H) is the compact operators and R(T) w is the weak closure of R(T). Weber’s result has been obtained by the author [3] as consequence of a more general result. A reasonable question is the following: Does Weber’s result holds when S = T = δA ∗ : B(H) 7 → B(H);X 7 → A∗X − XA∗? This question has been conjectured by the author in [5]. In this talk we give a positive answer to this conjecture. Other related results are also given