# On the Operator-valued $\mu$-cosine functions

## TL;DR

This paper characterizes hermitian operator-valued $$-cosine functions on topological abelian groups, showing they are expressed via continuous multiplicative operators, with applications to positive definite kernels and bounded cosine operators.

## Contribution

It provides a complete characterization of hermitian operator-valued -cosine functions, linking them to continuous multiplicative operators and extending previous results.

## Key findings

- Hermitian operator-valued -cosine functions have a specific form involving multiplicative operators.
- Explicit solutions of the cosine equation are derived using positive definite kernel theory.
- Connections to bounded normal cosine operators are established.

## Abstract

Let $(G,+)$ be a topological abelian group with a neutral element $e$ and let $\mu : G\longrightarrow\mathbb{C}$ 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 $ \Phi: G\longrightarrow \mathbf{B}(\mathcal{H})$ will be called an operator-valued $\mu$-cosine function if it satisfies both the $\mu$-cosine equation $$\Phi(x+y)+\mu(y)\Phi(x-y)=2\Phi(x)\Phi(y),\; x,y\in G$$ and the condition $\Phi(e)=I,$ where $I$ is the identity of $\mathbf{B}(\mathcal{H})$. We show that any hermitian operator-valued $\mu$-cosine functions has the form $$\Phi(x)=\frac{\Gamma(x)+\mu(x)\Gamma(-x)}{2}$$ where $ \Gamma: 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.