Let \(H\) be the Hilbert transform on the real line, \[ Hf(x)= {1\over\pi} \text{p.v. }\int^\infty_{-\infty} f(x- y) {dy\over y}\quad\text{for }x\in\mathbb{R}. \] Let \(T\) be a bounded linear operator on \(L^2(\mathbb{R})\) satisfying: (i) \(T\) maps real valued functions into real valued functions, (ii) \(T\) commutes with translations, (iii) \(-T^2= I\), the identity operator. Let \(A\) be the space of functions \(F= f+ iTf\), where \(f\in L^2(\mathbb{R})\) is real valued. Theorem 1. If \(A\) has the property that \(F^2\in A\) whenever \(F\in A\) and \(f^2\in L^2\), then \(T= \pm H\) and \(A= H^2\). The theorem has analogues if we replace \(\mathbb{R}\) with \(S^1\) or \(\mathbb{Z}\). Theorem 2. (a) Let \(T\) be a bounded linear operator on \(L^2(S^1)\) that satisfies (i)--(iii) and let \(A\) be the linear space of functions \(F= f+iTf\) with \(f\in L^2(S^1)\), real valued, such that \(\widehat f(0)= 0\). Suppose that \(A\) enjoys the same hypothesis as in Theorem 1. Then \(T= \pm H\), where \(H\) is now the conjugate function operator. (b) Let \(T\) be a bounded linear operator on \(L^2(\mathbb{Z})\) that satisfies (i)--(iii) and let \(A\) be the linear space of functions \(F= f+iTf\) with \(f\in L^2(\mathbb{Z})\), real valued. Then \(A\) cannot satisfy the same hypothesis as in Theorem 1. Namely, there exists \(F\in A\) such that \(f^2\in L^2(\mathbb{Z})\), but \(F^2\not\in A\).