The Hilbert-space adjoint of a bounded operator

Assume the Axiom of Countable Choice. Let $H$ and $K$ be real or complex Hilbert spaces and let $T\in \mathcal{B}(H,K)$ be a bounded linear operator (A bounded linear operator between normed spaces, The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators). For fixed $y\in K$ the map

$x⟼⟨Tx,y{⟩}_K$

is a bounded linear functional on $H$: it is linear in $x$ because $T$ is linear and the pairing is linear in its first argument, and $\mathrm{\mid }⟨Tx,y{⟩}_K\mathrm{\mid }\le \mathrm{\parallel }Tx\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }\le \mathrm{\parallel }T\mathrm{\parallel }\mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$ by Cauchy–Schwarz and the operator-norm inequality (Cauchy–Schwarz: $\mathrm{\mid }⟨x,y⟩\mathrm{\mid }\le \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$, with equality exactly for dependent pairs, The operator norm as the least bound and as the unit-sphere or unit-ball supremum). By Riesz representation (Riesz representation for Hilbert spaces) there is therefore a unique vector $T^{\ast }y\in H$ with

$⟨Tx,y{⟩}_K=⟨x,T^{\ast }y{⟩}_H\text{for every }x\in H\mathrm{.}$

The Hilbert adjoint of $T$ is the map

$T^{\ast }:K⟶H,y⟼T^{\ast }y\mathrm{.}$

It is the unique map $K\to H$ satisfying the displayed identity, since two such maps have $⟨x,(T_1^{\ast }-T_2^{\ast })y⟩=0$ for all $x$ and hence $(T_1^{\ast }-T_2^{\ast })y=0$ by positive definiteness.

The dictionary with the Banach transpose. Write $R_H:H\to H^{\ast }$ and $R_K:K\to K^{\ast }$ for the Riesz maps $R_H(x)=⟨\cdot ,x{⟩}_H$ and $R_K(y)=⟨\cdot ,y{⟩}_K$, and let $T^{\mathrm{\prime }}\in \mathcal{B}(K^{\ast },H^{\ast })$ be the transpose of $T$, $(T^{\mathrm{\prime }}g)(x)=g(Tx)$ (The transpose of a bounded operator). Then for all $y\in K$ and $x\in H$,

$(R_H(T^{\ast }y))(x)=⟨x,T^{\ast }y{⟩}_H=⟨Tx,y{⟩}_K=(R_{K}y)(Tx)=(T^{\mathrm{\prime }}(R_{K}y))(x),$

so $R_H{T}^{\ast }=T^{\mathrm{\prime }}{R}_K$. The Hilbert adjoint is thus the Banach transpose conjugated by the Riesz identifications of $H$ and $K$ with their duals; it is a different operator from $T^{\mathrm{\prime }}$ whenever the Riesz maps are conjugate-linear.