Norms of derivations of ℒ(X)

Barry E. Johnson

Vol. 38 (1971), No. 2, 465–469

DOI: 10.2140/pjm.1971.38.465

Abstract

If X is a real or complex Banach space and ℒ(X) is the algebra of bounded linear endomorphisms of X then each element T of ℒ(X) defines an operator D_T on ℒ(X) by D_T(A) = AT − TA. Clearly ∥D_T∥≦ 2inf _λ∥T + λI∥ and Stampfli has shown that when χ is a complex Hilbert space equality holds. In this paper it is shown, by methods which apply to a large class of uniformly convex spaces, that this formula for ∥D_T∥ is false in l^p and L^p(0,1)1 < p < ∞,p≠2. For L¹ spaces the formula is true in the real case but not in the complex case when the space has dimension 3 or more.

Mathematical Subject Classification 2000

Primary: 47B47

Milestones

Received: 2 November 1970

Published: 1 August 1971