Linear Operator

A linear Operator is a map $\Omega :V\to V$ satisfying

1. $\Omega a|V⟩=a\Omega |V⟩$
2. $\Omega (a|V⟩+b|W⟩)=a\Omega |V⟩+b\Omega |W⟩$
3. $⟨V|a\Omega =⟨V|\Omega a$
4. $(⟨V|a+⟨W|b)\Omega =a⟨V|\Omega +b⟨W|\Omega$

In the above:

• $V,W$ are vectors
• $a,b$ are scalars in a field
• $⟨\cdot |$ and $|\cdot ⟩$ follow the braket notation

Operators are a generalisation of matrices: They act on Vector Spaces linearly, without being attached to any specific basis. When working with finite Vector Spaces, the choice of a basis allows us to represent the Operator as a matrix. This won’t be always possible: Operators in infinite dimensional spaces do not always have a matrix representation

Relations

The diagram below aims to make relations between Operators and their matrix representation memorable. An arrow means an implication.

• Improve the relations below: They are lacking at the moment.

stateDiagram
	Hermitian: Hermitian
	Positive: Positive
	Bounded: Bounded Operators
	Real: Has real valued eigenvalues
	SelfAdjoint: Self Adjoint
	Spectral: Has a spectral decomposition
	Symmetric: Symmetric
	Unitary: Unitary
	Symmetric --> Hermitian : If Bounded
	Bounded --> Hermitian : If Symmetric
	Positive --> Spectral : If compact
	Hermitian --> Real
	SelfAdjoint --> Symmetric

See more details on:

• Adjoint Operator
• Hermitian Operator
• Anti-Hermitian Operator
• Self-Adjoint Operator
• Unitary Operator
• Symmetric Operator
• Involuntory Operator
• Compact Operator

Properties

• All operators can be decomposed into hermitian and anti-hermitian components
• In finite dimensional spaces, hermitian = symmetric = self-adjoint