Integral transforms of the Kontorovich-Lebedev convolution type

We deal with a class of integral transformations of the form \begin{flushleft}$f(x) \rightarrow\frac{1}{2x}\prod_{n=1}^\infty(1+\frac{x(x-\frac{d}{dx}-\frac{d^2} {dx^2})}{(2n-1)^2})\int_{\mathbb{R}_+^2}e^{-{\frac{1}{2} (x\frac{u^2+y^2} {uy}+ \frac{yu}{x})}_{f(u)h(y)dudy,x\in\mathbb{R}+}}$ \end{flushleft}in $L_2(\mathbb {R}_+;xdx)$, which is associated with the Kontorovich-Lebedev operator \begin{center} $K_{i\tau}[f]=\int_0^\infty K_{i\tau}(x)f(x)dx,\tau\in\mathbb{R}+$.
\end{center} Necessary and sufficient conditions on h to establish that the transformation is unitary in $L_2(\mathbb{R}+;xdx)$ are obtained. A reciprocal inversion formula and an example of the unitary convolution transformation are given.
given.