Derivations on symmetric quasi-Banach ideals of compact operators
Abstract
Let
I,J
be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space
H
, let
J:I
be a 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
∥
B(H)
≤∥δ
∥
I→J
≤2C∥a
∥
J:I
, where
C
is the modulus of concavity of the quasi-norm
∥⋅
∥
J
. 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)
.