Counterexample to the Trotter product formula for projections

Abstract

We constructed a unitary semigroup $(e^{tA}{)}_{t\ge 0}$ on a Hilbert space and an orthogonal projection $P$ such that the limit ${\lim }_{n\to \infty }[e^{\frac{t}{n}A}P{]}^n$ does not exist strongly. A similar example with a positive contractive semigroup and positive contractive projection on $L_p$ is also constructed.