I have read that there are two 'options' for an adjoint when dealing with Hilbert spaces.
Let $T : X \to Y$ be a bounded linear operator between the Hilbert spaces $X$ and $Y$.
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The Hilbert space adjoint: Define $T^* : Y \to X$ by $( T^*(y), x)_X = (y, T x)_Y$.
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The "usual" Banach space adjoint: Define $T^* : Y' \to X'$ by $\langle T^*(y^*), x \rangle_{X',X} = \langle y^*, T x\rangle_{Y',Y}$.
Here, $(\cdot,\cdot)_X$ refers to the scalar product in $X$, whereas $\langle \cdot, \cdot \rangle_{X',X}$ refers to the duality product between $X'$ and $X$.
It seems to me that these are both exactly the same, its just that we identify the duality product with the scalar product because we are in a Hilbert space and have an inner product at our disposal. Is this correct or are the adjoints in 1. and 2. fundamentally different?
Best Answer
$\newcommand{\hdual}{\mathsf H}$Let $T^\hdual$ denote the "Hilbertian" adjoint (usually called conjugate transpose) and $T^*$ the "Dual" or "Banachian" adjoint. Let $\mathrm E_X:X \to X^*$ be the natural embedding of a Hilbert space onto its dual, namely $x \mapsto \langle x,\cdot \rangle$. Here the inner product is taken to be linear in the second component and conjugate-linear in the first.
We can show that $$ T^\hdual = \mathrm E_X^{-1} T^* \mathrm E_Y^{\vphantom{-1}} $$