If $T$ is a densely defined operator on a Hilbert space $\mathbf{H}$, then one can mimic the Riesz representation theorem to define an adjoint operator for $T$ whose domain is given by all those elements $x \in \mathbf{H}$ such that the map $$ \text{Dom}(T) \to \mathbb{C}, ~~~~ y \mapsto \langle x, T(y)\rangle $$ is bounded (where the inner product is linear in the first variable). The domain can of course be $0$, however, the adjoint always exists.

In the literature on bounded operators on Hilbert spaces (and Hilbert modules) however, one often speaks of an adjointable operator. Am I correct in guessing that the this precisely means that the adjoint of the operator is defined everywhere. Since otherwise the adjective would be trivial.



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