Strictly singular and compact operators

Abstract:

Given Banach spaces E and F, a linear operator A : E → F is called strictly singular if the restriction of A to any infinite–dimensiinal subspace of E is not an isomorphism. This consept was introduced by T. Kato. Strictly singular operators form an operator ideal which contains the ideal of compact operators. We will denote by SS(Lp) (K(Lp)) the set of strictly singular (compact) operators in Lp = Lp[0, 1]. It is well known that SS(Lp) = K(Lp) iff p = 2. Theorem 1. Let 1 < q < r < ∞ and A be a linear operator bounded in Lq and Lr. Then one of the following alternatives holds: 1. A ∈ K(Lp) for any p ∈ (q, r), or 2. A∈SS(Lp) for any p ∈ (q, r). Let 1 6 p 6 ∞. Given a bounded linear operator A in Lp, denote O(A) = {q : 1 6 q 6 ∞, A is bounded in Lq}. By M. Riesz interpolation theorem O(A) is a convex subset R1 . This subset may be open, closed or semiclosed. Denote Vp = SS(Lp) \ K(Lp). Theorem 2. Let 1 < p < ∞, p 6= 2 and A ∈ Vp. Then p is endpoint of O(A). Moreover, if p > 2, then p is the right endpoint of O(A), if p < 2, then p is the left endpoint of O(A). All known examples of operators in Vp always depend on p. The following theorem explains this phenomenon. Theorem 3. Let 1 < q < p < ∞. The set Vp∩Vq is not empty iff q < 2 < p. The author was partly supported by RFBR, grant 08–01–00226, and Complutence University Joint work with F. L. Hernandez and P. Tradacete.