3 KiB
Operator classes
Hilbert-adjoint operator
Definition 1: let
(X, \langle \cdot, \cdot \rangle_X)
and(Y, \langle \cdot, \cdot \rangle_Y)
be Hilbert spaces over the fieldF
and letT: X \to Y
be a bounded linear operator. The Hilbert-adjoint operatorT^*
ofT
is the operatorT^*: Y \to X
such that for allx \in X
amdy \in Y
\langle Tx, y \rangle_Y = \langle x, T^* y \rangle.
We should first prove that for a given T
such a T^*
exists.
Proposition 1: the Hilbert-adjoint operator
T^*
ofT
exists is unique and is a bounded linear operator with norm
|T^*| = |T|.
??? note "Proof:"
Will be added later.
The Hilbert-adjoint operator has the following properties.
Proposition 2: let
T,S: X \to Y
be bounded linear operators, then
\forall x \in X, y \in Y: \langle T^* y, x \rangle_X = \langle y, Tx \rangle_Y
,(S + T)^* = S^* + T^*
,\forall \alpha \in F: (\alpha T)^* = \overline \alpha T^*
,(T^*)^* = T
,\|T^* T\| = \|T T^*\| = \|T\|^2
,T^*T = 0 \iff T = 0
,(ST)^* = T^* S^*, \text{ when } X = Y
.
??? note "Proof:"
Will be added later.
Self-adjoint operator
Definition 2: a bounded linear operator
T: X \to X
on a Hilbert spaceX
is self-adjoint if
T^* = T.
If a basis for \mathbb{C}^n
(n \in \mathbb{N})
is given and a linear operator on \mathbb{C}^n
is represented by a matrix, then its Hilbert-adjoint operator is represented by the complex conjugate transpose of that matrix (the Hermitian).
Proposition 3, 4 and 5 pose some interesting results of self-adjoint operators.
Proposition 3: let
T: X \to X
be a bounded linear operator on a Hilbert space(X, \langle \cdot, \cdot \rangle_X)
over the field\mathbb{C}
, then
T \text{ is self-adjoint} \iff \forall x \in X: \langle Tx, x \rangle \in \mathbb{R}.
??? note "Proof:"
Will be added later.
Proposition 4: the product of two bounded self-adjoint linear operators
T
andS
on a Hilbert space is self-adjoint if and only if
ST = TS.
??? note "Proof:"
Will be added later.
Commuting operators therefore imply self-adjointness.
Proposition 5: let
(T_n)_{n \in \mathbb{N}}
be a sequence of bounded self-adjoint operatorsT_n: X \to X
on a Hilbert spaceX
. IfT_n \to T
asn \to \infty
, thenT
is a bounded self-adjoint linear operator onX
.
??? note "Proof:"
Will be added later.
Unitary operator
Definition 3: a bounded linear operator
T: X \to X
on a Hilbert spaceX
is unitary ifT
is bijective andT^* = T^{-1}
.
A bounded unitary linear operator has the following properties.
Proposition 6: let
U, V: X \to X
be bounded unitary linear operators on a Hilbert spaceX
, then
U
is isometric,\|U\| = 1 \text{ if } X \neq \{0\}
,UV
is unitary,U
is normal, that isU U^* = U^* U
,T \in \mathscr{B}(X,X)
is unitary\iff
T
is isometric and surjective.
??? note "Proof:"
Will be added later.