Vol. 38, No. 2, 1971

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Norms of derivations of (X)

Barry E. Johnson

Vol. 38 (1971), No. 2, 465–469
Abstract

If X is a real or complex Banach space and (X) is the algebra of bounded linear endomorphisms of X then each element T of (X) defines an operator DT on (X) by DT(A) = AT TA. Clearly DT2inf λT + λIand 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 DTis false in lp and Lp(0,1)1 < p < ,p2. 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
Authors
Barry E. Johnson