# Adjoint for a non-densely defined unbounded operator on a Hilbert space

Let $$\mathbf{H}$$ be a Hilbert space, and $$D$$ an unbounded densely-defined operator on $$\mathbf{H}$$. As is well-known, every such operator admits an adjoint, with domain possibly different from that of $$D$$.

Is it possible to define an adjoint for a non-densely defined operator?

• Are you asking about how the usual definition breaks down in this case, or are you asking about possible alternate definitions that might work instead? – Nate Eldredge Mar 4 at 21:10
• I'm asking about alternate definitions – Max Schattman Mar 4 at 22:05
• I guess one sort of trivial approach is to consider the operator $DP$ where $P$ is orthogonal projection onto the closure of $\operatorname{dom D}$. In other words, extend your operator by zero on $(\operatorname{dom} D)^\perp$. Now $DP$ is densely defined and you may consider its adjoint in the usual sense, which formally should equal $PD^*$. – Nate Eldredge Mar 5 at 1:51
• The issue when $D$ is not densely defined is that, since $D^* x$ is only defined via its inner product with elements of $\operatorname{dom} D$, it is only well defined up to adding something from $(\operatorname{dom} D)^\perp$. But there is a "natural" choice of what to add, namely 0. – Nate Eldredge Mar 5 at 1:58