A selfadjoint operator T ≠ 0 on a Hilbert space such that T is neither positive nor negative.

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Consider T = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) belonging to B(C^{2}), ξ = ( \begin{matrix}1 \\ 0 \end{matrix} ) and η = ( \begin{matrix}0 \\ 1 \end{matrix} ). Then < T ξ, ξ >= −1 and < T η, η >= 1. Hence the selfadjoint operator T is neither positive nor negative.