notes/docs/mathematics/functional-analysis/inner-product-spaces/operator-classes.md

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 field F and let T: X \to Y be a bounded linear operator. The Hilbert-adjoint operator T^* of T is the operator T^*: Y \to X such that for all x \in X amd y \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^* of T exists is unique and is a bounded linear operator with norm

|T^*| = |T|.

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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

1. \forall x \in X, y \in Y: \langle T^* y, x \rangle_X = \langle y, Tx \rangle_Y,
2. (S + T)^* = S^* + T^*,
3. \forall \alpha \in F: (\alpha T)^* = \overline \alpha T^*,
4. (T^*)^* = T,
5. \|T^* T\| = \|T T^*\| = \|T\|^2,
6. T^*T = 0 \iff T = 0,
7. (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 space X 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 and S on a Hilbert space is self-adjoint if and only if

ST = TS.

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Commuting operators therefore imply self-adjointness.

Proposition 5: let (T_n)_{n \in \mathbb{N}} be a sequence of bounded self-adjoint operators T_n: X \to X on a Hilbert space X. If T_n \to T as n \to \infty, then T is a bounded self-adjoint linear operator on X.

??? note "Proof:"

Will be added later.

Unitary operator

Definition 3: a bounded linear operator T: X \to X on a Hilbert space X is unitary if T is bijective and T^* = 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 space X, then

1. U is isometric,
2. \|U\| = 1 \text{ if } X \neq \{0\},
3. UV is unitary,
4. U is normal, that is U U^* = U^* U,
5. T \in \mathscr{B}(X,X) is unitary \iff T is isometric and surjective.

??? note "Proof:"

Will be added later.