Linear Transformation is Isomorphism iff Inverse Equals Adjoint

Theorem
Let $H, K$ be Hilbert spaces.

Let $U \in B \struct {H, K}$ be a bounded linear transformation.

Then the following are equivalent:


 * $(1): \quad U$ is an isomorphism
 * $(2): \quad U$ is invertible and $U^{-1} = U^*$, where $U^*$ denotes the adjoint of $U$.