Yes, there is one such example: $u \equiv 0$. 

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The answer above is not facetious! That $u$ is in fact the only example (modulo measure zero modifications). 

By [Titchmarsch's theorem](https://en.wikipedia.org/wiki/Hilbert_transform#Titchmarsh.27s_theorem), if $u\in L^2(\mathbb{R})$ and its Fourier support is on the positive real line, $u$ must be equal to the trace of some holomorphic function $F$ defined on the upper half plane. 

If $u$ itself further vanishes on the left half line, which has positive measure, by the [Luzin-Privalov Theorem](https://www.encyclopediaofmath.org/index.php/Luzin%E2%80%93Privalov_theorems) the function $F$ must vanish identically. Hence the only function satisfying your condition is identically zero.