The infinite-order integro-differential operator related to the Lebedev–Skalskaya transform

In this article, we introduce infinite-order integro-differential operator related to Lebedev–Skalskaya transform. Some characteristics of this operator are obtained. Furthermore, we establish the necessary and sufficient conditions for a class of infinite-order integro-differential operators to be unitary on \( L^2({\mathbb {R}}_{+}; \, dx)\) $L^2({\mathbb{R}}_{+};dx)$ . Some classes of related integro-differential equations are also studied at the end.