# Product formulas in the framework of mean ergodic theorems

## TL;DR

This paper extends Chernoff's product formula for operator-valued functions, with applications in quantum dynamical control, including decoupling and the Quantum Zeno effect, by analyzing iterates of contractions on Banach spaces.

## Contribution

It introduces a generalized product formula for one-parameter operator functions, linking it to quantum control techniques.

## Key findings

- Extended Chernoff's product formula for Banach space operators.
- Connected the product formula to quantum dynamical control methods.
- Provided mathematical framework for quantum Zeno effect applications.

## Abstract

An extension of Chernoff's product formula for one-parameter functions taking values in the space of bounded linear operators on a Banach space is given. Essentially, the $n$-th one-parameter function in the product formula is mapped by the $n$-th iterate of a contraction acting on the space of the one-parameter functions. The motivation to study this specific product formula lies in the growing field of dynamical control of quantum systems, involving the procedure of dynamical decoupling and also the Quantum Zeno effect.