Abstract: In this paper, we introduce and study a bicomplex analog of slant Hankel operators acting on the bicomplex Hilbert space $$L_{\mathbb{BC}}^{2}(U_{\mathbb{BC}}).$$ Motivated by the classical theory of slant Hankel operators defined on $$L^{2}(\mathbb{T}),$$ we extend the underlying framework to the setting of bicomplex-valued functions. For a symbol $$\varphi=\varphi_{1}e_{1}+\varphi_{2}e_{2}$$ in $$L_{\mathbb{BC}}^{\infty},$$ we define the associated slant bicomplex Hankel operator via its matrix representation with respect to the standard orthonormal basis, preserving the characteristic slant structure. We investigate fundamental operator-theoretic properties of these operators, including boundedness and norm estimates, and establish conditions for an operator to be a slant bicomplex Hankel operator. Utilizing the idempotent decomposition of bicomplex numbers, we derive representations that allow the reduction of certain problems to classical complex components.

Keywords: Bicomplex numbers, Bicomplex Hilbert space, Idempotent decomposition, Hyperbolic norm, Slant Hankel operator, Toeplitz operator

Pages: 376-385