Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ Can $T$ be decomposed as $T = S^{\ast} S$ for a bounded operator on $S$ on $L^2$, where $S^{\ast}$ denotes the adjoint of $S$?
Particular measures of interest are the natural measure on the sphere $S^{n-1}$ and the natural measure on the middle thirds Cantor set (when $n=1$).
If $\widehat{\mu}$ is non-negative, then its square root defines a bounded Fourier multiplier operator $S$ on $L^2$. In this case $S$ is self-adjoint, so $T=S^*S$.
Conversely, any linear and shift invariant $S$ is a Fourier multiplier operator with an $L^\infty$ symbol. The adjoint $S^*$ would have a symbol that is the complex conjugate of the symbol for $S$. Therefore the symbol of $S^*S$ must be non-negative.