Derivations on symmetric quasi-Banach ideals of compact operators

Let I,JI,J be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space HH, let J : IJ : I be the space of multipliers from II to JJ. Obviously, ideals II and JJ are quasi-Banach algebras and it is clear that ideal JJ is a bimodule for II. We study the set of all derivations from II into JJ. We show that any such derivation is automatically continuous and there exists an operator a∈J : Ia∈J : I such that δ(⋅)=[a,⋅]δ(⋅)=[a,⋅], moreover ‖a+α1‖B(H)≤‖δ‖I→J≤2C‖a‖J:I‖a+α1‖B(H)≤‖δ‖I→J≤2C‖a‖J:I for some complex number αα, where CC is the modulus of concavity of the quasi-norm ‖⋅‖J‖⋅‖J and 11 is the identity operator on HH. In the special case, when I=J=K(H)I=J=K(H) is a symmetric Banach ideal of compact operators on HH our result yields the classical fact that any derivation δδ on K(H)K(H) may be written as δ(⋅)=[a,⋅]δ(⋅)=[a,⋅], where aa is some bounded operator on HH and ‖a‖B(H)≤‖δ‖I→I≤2‖a‖B(H)‖a‖B(H)≤‖δ‖I→I≤2‖a‖B(H).