Isomorphism iff Inverse Equals Adjoint

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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 $U$.


Proof


Sources