## Operators

A **linear map** between two vector spaces \mathcal{H} and \mathcal{K} is a function A\colon\mathcal{H}\to\mathcal{K} that respects linear combinations:
A(c_1|v_1\rangle+c_2|v_2\rangle)=c_1 A|v_1\rangle+c_2 A|v_2\rangle
for any vectors |v_1\rangle,|v_2\rangle and any complex numbers c_1,c_2.
We will focus mostly on **endomorphisms**, that is, maps from \mathcal{H} to \mathcal{H}, and we will call them **operators**.
The symbol \mathbf{1} is reserved for the identity operator that maps every element of \mathcal{H} to itself (i.e. \mathbf{1}|v\rangle=|v\rangle for all |v\rangle\in\mathcal{H}).
The product BA of two operators A and B is the operator obtained by first applying A to some ket |v\rangle and then B to the ket which results from applying A:
(BA)|v\rangle = B(A|v\rangle).
The order *does* matter: in general, BA\neq AB.
In the exceptional case in which AB=BA, one says that these two operators **commute**.
The inverse of A, written as A^{-1}, is the operator that satisfies AA^{-1}=\mathbf{1}=A^{-1}A.
For finite-dimensional spaces, one only needs to check *one* of these two conditions, since any one of the two implies the other, whereas, on an infinite-dimensional space, *both* must be checked.
Finally, given a particular basis, an operator A is uniquely determined by the entries of its matrix: A_{ij}=\langle i|A|j\rangle.

The **adjoint**, or **Hermitian conjugate**, of an linear map A, denoted by A^\dagger, is defined by the relation
\begin{gathered}
\langle i|A^\dagger|j\rangle
= \langle j|A|i\rangle^\star
\\\text{for all $|i\rangle,|j\rangle\in\mathcal{H}$}
\end{gathered}
and \dagger turns (n\times m) matrices into (m\times n) matrices.

An operator A is said to be

**normal** if AA^\dagger = A^\dagger A
**unitary** if A^\dagger=A^{-1}
**Hermitian** (or **self-adjoint**) if A^\dagger = A.

In particular then, being unitary implies being normal, since if A^\dagger=A^{-1} then AA^\dagger=A^\dagger A, since both of these are equal to \mathbf{1}.
Note also that unitary and Hermitian operators must be operators, i.e. they are represented by an (n\times n) matrix.

Any *physically admissible* evolution of an isolated quantum system is represented by a unitary operator.
Note that unitary operators preserve the inner product: given a unitary operator U and two kets |a\rangle and |b\rangle, and defining |a'\rangle=U|a\rangle and |b'\rangle=U|b\rangle, we have that
\begin{gathered}
\langle a'|=\langle a|U^\dagger
\\\langle b'|=\langle b|U^\dagger
\\\langle a'|b'\rangle=\langle a|U^\dagger U|b\rangle=\langle a|\mathbf{1}|b\rangle=\langle a|b\rangle.
\end{gathered}
Preserving the inner product implies preserving the norm induced by this product, i.e. unit state vectors are mapped to unit state vectors, i.e. *unitary operations are the isometries of the Euclidean norm*.