On the Operator-valued $μ$-cosine functions

Abstract

Let $(G,+)$ be a topological abelian group with a neutral element $e$ and let $μ: G\longrightarrow\mathbb{C}$ be a continuous character of $G$. Let $(\mathcal{H}, \langle \cdot,\cdot \rangle)$ be a complex Hilbert space and let $\mathbf{B}(\mathcal{H})$ be the algebra of all linear continuous operators of $\mathcal{H}$ into itself. A continuous mapping $ Φ: G\longrightarrow \mathbf{B}(\mathcal{H})$ will be called an operator-valued $μ$-cosine function if it satisfies both the $μ$-cosine equation $$Φ(x+y)+μ(y)Φ(x-y)=2Φ(x)Φ(y),\; x,y\in G$$ and the condition $Φ(e)=I,$ where $I$ is the identity of $\mathbf{B}(\mathcal{H})$. We show that any hermitian operator-valued $μ$-cosine functions has the form $$Φ(x)=\frac{Γ(x)+μ(x)Γ(-x)}{2}$$ where $ Γ: G\longrightarrow \mathbf{B}(\mathcal{H})$ is a continuous multiplicative operator. As an application, positive definite kernel theory and W. Chojnacki's results on the uniformly bounded normal cosine operator are used to give explicit formula of solutions of the cosine equation.