Existence and Uniqueness of the Adjoint Operator

Let $H$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-hilbert-space-2026a}{complex Hilbert space}, with \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$, which is a \reftext{def:complex-normed-space-2026a}{norm} on $H$ by claim 2 of \ref{lem:inner-product-norm-is-norm-2026a}, and let $T$ be a \reftext{def:bounded-linear-operator-2026a}{bounded linear operator} on $H$. Adjoints are as in \reftext{def:adjoint-operator-2026a}{that definition}, and sums of vectors are \reftext{def:finite-sum-vector-space-2026a}{finite sums in $H$}. Then the following hold.

\textbf{1. (Uniqueness)} $T$ has at most one adjoint. When an adjoint of $T$ exists we write $T^{*}$ for it.

\textbf{2. (Existence given a finite orthonormal basis)} Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $e:[n]\to H$ be an \reftext{def:orthonormal-basis-2026a}{orthonormal basis} of $H$. Then the map sending $u\in H$ to

is a bounded linear operator on $H$ and is an adjoint of $T$. In particular $T$ has an adjoint $T^{*}$, and $T^{*}(u)$ is given by the displayed formula for every $u\in H$; since the condition in \reftext{def:adjoint-operator-2026a}{the definition of an adjoint} does not mention $e$, the same operator arises from every orthonormal basis of $H$.