 Solution

Let $d=\mu -(1+r)P_0$. Since $V$ is symmetric, $A=\mathrm{Im}V=(\ker V{)}^{\perp }$. If $d\in A$, every $H\in \ker V$ satisfiesAny $H\notin \ker V$ has a nondegenerate normal terminal value and cannot be nonnegative almost surely. Any $H\in \ker V$ has the displayed deterministic relation, which excludes an arbitrage because $1+r=(\eta \cdot \mu )/(\eta \cdot P_0)>0$.

Conversely, if $d\notin A$, choose $h\in \ker V$ with $h\cdot d>0$. The zero-cost portfoliohas deterministic terminal payoffIt is a terminal-consumption arbitrage. Thus no arbitrage is equivalent to $d\in A$, the Arbitrage in a one-period Gaussian market criterion.