On the Range Inclusion of Normal Derivations: Variations on a Theme by Johnson, Williams and Fong

This paper considers various extensions of results of Johnson and Williams and of Fong on the range inclusion of normal derivations. Let C p , with p ∈ [1, ∞), be the Schatten ideals of compact operators on a Hilbert space H with norms | | p , C ∞ be the ideal of all compact operators, and C b the algebra B ( H ) of all bounded operators. Any S ∈ B ( H ) defines a bounded derivation δ S on all C p : δ S ( X ) = SX − XS , for X ∈ C p . Johnson and Williams proved that, for normal S , the inclusion of ranges δ T ( C b ) ⊆ δ S ( C b ) implies T = g ( S ), where g is a Lipschitz, differentiable function on σ( S ). Fong showed that the condition T = g ( S ) is equivalent to the range inclusion δ T ( C 1 ) ⊆ δ S ( C b ). This paper studies the range inclusionδ T ( C p ) ⊆ δ S ( C p ) for normal S and p ∈ [1, ∞] ∪ b , and the classes of functions for which T = g ( S ). Setp − = min ( p , p p − 1 ) ,p + = max ( p , p p − 1 ) , for  p ∈ ( 1 , ∞ ) , andp − = 1 ,p + = b , for  p ∈ { 1 , ∞ , b } . This paper shows that condition (1) implies: (i) the range inclusions δ T ( C r ) ⊆ δ S ( C r ), for r ∈ [ p − , p + ]; (ii) that there exists D > 0 such that |δ T ( X )| p ⩽ D |δ S ( X )| p , for X ∈ B ( H ) (| X | p = ∞ if X ∉ C p ); (iii) the range inclusions δ g ( A )( C p ) ⊆ δ A ( C p ) for any normal operator A with σ( A ) ⊆ σ( S ). It establishes that (1) implies that the function g (in T = g ( S )) is C p ‐Lipschitzian on σ( S ), that is, there is D > 0 such that | g ( A ) − g ( B )| p ⩽ D | A − B | p for all normal A and B with spectra in σ( S ). Conversely, it is proved that, for any selfadjoint S and C p ‐Lipschitz function g on σ( S ), δ g ( S )( C p ) ⊆ δ S ( C p ). The paper also extends the above results of Johnson and Williams to bounded derivations of C *‐algebras. 2000 Mathematics Subject Classification : 46L57, 47B47, 58C07.