Hermitian Adjoint - Properties

Properties

Immediate properties:

  1. A** = A
  2. If A is invertible, then so is A*, with (A*)−1 = (A−1)*
  3. (A + B)* = A* + B*
  4. (λA)* = λ* A*, where λ* denotes the complex conjugate of the complex number λ
  5. (AB)* = B*A*

If we define the operator norm of A by

then

.

Moreover,

The set of bounded linear operators on a Hilbert space H together with the adjoint operation and the operator norm form the prototype of a C* algebra.

The relationship between the image of and the kernel of its adjoint is given by:

Proof of the first equation:

\begin{align} A^* x = 0 &\iff \langle A^*x,y \rangle = 0 \quad \forall y \in H \\ &\iff \langle x,Ay \rangle = 0 \quad \forall y \in H \\ &\iff x\ \bot \ \operatorname{im}\ A \end{align}

The second equation follows from the first by taking the orthogonal space on both sides. Note that in general, the image need not be closed, but the kernel of a continuous operator always is.

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