Let \(H_1\) and \(H_2\) be Hilbert spaces and let \(T\colon H_1\to H_2\) denote a bounded operator . The adjoint of T is the continuous operator \(T^*\colon H_2\to H_1\) satisfying

\[ \langle Tx, y\rangle_{H_2} = \langle x, T^*y\rangle_{H_1} \]

for all \(x\in H_1\) and \(y\in H_2\).

Remarks

  • For every \(y\in H_2\), the map

    \[ x \mapsto \langle Tx, y\rangle_{H_2} \]

    is bounded and linear. By the Riesz representation theorem , there exists a vector \(z\in H_1\) such that

    \[ \langle Tx, y\rangle_{H_2} = \langle x, z\rangle_{H_1}. \]

    Then we set \(T^*y=z\). This construction is equivalent to the above definition.

  • Properties of Adjoint Operators. Let \(H_1\), \(H_2\), and \(H_3\) denote Hilbert spaces, \( S, T \in \mathcal{L}(H_1, H_2) \), \( R \in \mathcal{L}(H_2, H_3) \), and \( \lambda \in \mathbb{C} \).

    1. \((S + T)^* = S^* + T^*\)
    2. \((\lambda S)^* = \overline{\lambda}\, S^*\)
    3. \((RS)^* = S^* R^*\)
    4. \(S^{**} = S\)
    5. \(S^* \in \mathcal{L}(H_2, H_1)\) and \(\|S^*\| = \|S\|\)
    6. \(\|S^* S\| = \|S S^*\| = \|S\|^2\)
    7. \(\ker S = (\operatorname{ran} S^*)^{\perp}_{H_1}\) (\(Sx = 0 \;\; \Leftrightarrow \;\; S^* Sx = 0 \quad (1.5,\; \text{Satz 26})\))