On the Bogolyubov endomorphisms of the renormalized square of white noise algebra

We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple [Formula: see text], where [Formula: see text] are linear transformations from a test-function space [Formula: see text] into itself, while [Formula: see text] is anti-linear on [Formula: see text] and [Formula: see text] is real. Precisely, we prove that [Formula: see text] and [Formula: see text] are uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text], into itself. Under some additional analytic conditions on [Formula: see text] and [Formula: see text], we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case [Formula: see text] and the other corresponds to the case [Formula: see text]. Finally, we close the paper by building some examples in one and multi-dimensional cases.