Convolution operator in $L^2(\mathbb{R})$

Prove that the convolution operator $S_{f}$ in $L_{2}(\mathbb {R})$ is unitarily equivalent to the operator of multiplication by the function $\sqrt{2 \pi } \hat{f}: \hat{F} S_{f} \hat{F}^{-1} h = \sqrt{2 \pi } \hat{f} h$ for every function $h \in L_{2}(\mathbb {R})$.

Find the spectrum of the convolution operator with the function $e^{-\left|x\right|}$ in $L_{2}(\mathbb {R})$.

Find the spectrum of the operator $A: L_{2}(\mathbb {R}) \rightarrow L_{2}(\mathbb {R})$ of the following form:

Is this operator compact?

Find the spectrum of the Hilbert operator in $L_{2}(\mathbb {R})$:

(N. Wiener, 1933). Let $f \in L_{2}(\mathbb {R})$. Prove that the system $\left\{ f(x-a): a \in \mathbb {R}\right\}$ of shifts of this function is complete in $L_{2}(\mathbb {R})$ if and only if the measure of the set $\left\{ y \in \mathbb {R}: \hat{f}(y) = 0\right\}$ equals zero.