A counterexample in the theory of Hermitian liftings

Abstract

In [3], [8], and [2], it was shown that if is an essentially Hermitian operator on l P, 1≦ p<∞, or on Lp[0,1], 1< p<∞, then T is a compact perturbation of a Hermitian operator. In [1], this result was established for operators on Orlicz sequence space l M, where 2∉[α M,β M] (the associated interval for M). In that same paper, it was conjectured that this result does not in general hold if 2∈[α M,β M]. In this paper, we show that this conjecture is correct by exhibiting an Orlicz sequence space l M and an essentially Hermitian operator on l M which is not a compact perturbation of a Hermitian operator.