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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

\[ \|T^*\| = \|T\|. \]
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

  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. \((ST)^* = T^* S^*, \text{ when } X = Y\).
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

\[ T \text{ is self-adjoint} \iff \forall x \in X: \langle Tx, x \rangle \in \mathbb{R}. \]
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

\[ ST = TS. \]
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 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 \(X\).

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.
Proof:

Will be added later.