Functional-Vector Spaces $\mathscr{L}^2 (X,σ(X),\mu_t)$ with Scaled-Hyperbolic-Valued Probabilities $\mu_t$

Abstract

In this paper, we construct a natural $\mathbb{D}_t$-valued-functional-vector space $$\mathscr{L}^2(X,σ(X),\mu_t)$$on a measurable space $(X,σ(X))$ up to a $\mathbb{D}_t$-valued probability $\mu_t,$ where $\mathbb{D}_t$ is the $t$-scaled hyperbolic numbers for an arbitrarily fixed scale $t ∈ \mathbb{R}.$ We show that this vector space $\mathscr{L}^2 (X,σ(X),\mu_t)$ is Banach space over the real field $\mathbb{R},$ equipped with symmetric $\mathbb{D}_t$-valued-positive $\mathbb{R}$-bilinear form for any $t ∈ R.$ As application, we consider multiplication operators acting on $\mathscr{L}^2 (X,σ(X),\mu_t).$