Properties
Immediate properties:
- A** = A
- If A is invertible, then so is A*, with (A*)−1 = (A−1)*
- (A + B)* = A* + B*
- (λA)* = λ* A*, where λ* denotes the complex conjugate of the complex number λ
- (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:
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.
Read more about this topic: Hermitian Adjoint
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)