They are a generalization of unitary matrix.
A unitary operator is a bounded linear operator $U: H \to H$ on a Hilbert space $H$ that preserves the inner product. In other words, for any vectors $x$ and $y$ in $H$, we have:
$$ \langle Ux, Uy \rangle = \langle x, y \rangle $$Another equivalent definition is that an operator $U$ is unitary if it is invertible and $U^* = U^{-1}$, where $U^*$ is the adjoint of $U$.
They have properties like preserving norms ($\| Ux \| = \| x \|$) and being "distance-preserving" in the Hilbert space.
They are in close relationship to self adjoint operators, given by the Stone's theorem. Roughly speaking, every 1-parameter subgroup of unitary operators is generated by a self adjoint operator (multiplied by $i$).
Two unitary operators commute if and only if their self adjoint operators commute: if two unitary operators $U_1$ and $U_2$ commute, meaning $U_1 U_2 = U_2 U_1$, then their generating self-adjoint operators $A_1$ and $A_2$ will also commute, $A_1 A_2 = A_2 A_1$.
Let's consider that the unitary operators are generated by self-adjoint operators as:
$$ U_1(t) = \exp(itA_1), \quad U_2(t) = \exp(itA_2). $$If $A_1$ and $A_2$ commute then
$$ \exp(itA_1)\exp(itA_2) = \exp(itA_1+itA_2), $$and
$$ \exp(itA_2)\exp(itA_1) = \exp(itA_1+itA_2), $$so $U_1(t),U_2(t)$ commute.
They play a fundamental role in quantum mechanics, where they often represent evolution though a parameter: time evolution, continuous rotations, and so on...
Indeed, they are the biggest set of accepted transformations of a quantum mechanical system (any other transformation would "break the stage" in which everything is happening, the corresponding Hilbert space). If we additionally require to the unitary transformations to preserve the specific Hamiltonian of the system then we obtain the symmetry group of a physical system.
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Author of the notes: Antonio J. Pan-Collantes
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