An example in the theory of -operators

-operators are a generalization in the context of well-boundedness of normal operators on Hilbert space. It was shown by Doust and Walden that compact -operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one suggested in their paper.