Derivations on symmetric quasi-Banach ideals of compact operators

Let I,J be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space H , let J : I be the space of multipliers from I to J . Obviously, ideals I and J are quasi-Banach algebras and it is clear that ideal J is a bimodule for I . We study the set of all derivations from I into J . We show that any such derivation is automatically continuous and there exists an operator a ∈ J : I such that δ ( ⋅ ) = [ a , ⋅ ] , moreover ‖ a + α 1 ‖ B ( H ) ≤ ‖ δ ‖ I → J ≤ 2 C ‖ a ‖ J : I for some complex number α , where C is the modulus of concavity of the quasi-norm ‖ ⋅ ‖ J and 1 is the identity operator on H . In the special case, when I = J = K ( H ) is a symmetric Banach ideal of compact operators on H our result yields the classical fact that any derivation δ on K ( H ) may be written as δ ( ⋅ ) = [ a , ⋅ ] , where a is some bounded operator on H and ‖ a ‖ B ( H ) ≤ ‖ δ ‖ I → I ≤ 2 ‖ a ‖ B ( H ) .