\label{a+a-n}$$ where the number operator $N$, by definition, satisfies the commutation relations $$[a^{-},N]=a^{-},\quad
[a^{+},N]=-a^{+},$$ and equality $N|0>=0$, and, as a consequence, $$N|n>=n|n>.$$ Using eqs. (\[a+a-n\]) and (\[hei\]), we get $$a^{-}a^{+}=N+1+\nu\Pi_{+},$$ and, as a result, we arrive at the following expression for the number operator in terms of the operators $a^{\pm}$: $$N=\frac{1}{2}\{a^{-},a^{+}\}-\frac{1}{2}(\nu+1).
\label{ndef}$$ Therefore, due to the completeness of the set of basis states $|n>$, we can realize the Klein operator $K$ in terms of the operators $a^{\pm}$ by means of equality (\[ndef\]): $
K=\exp{i\pi N},
$ or, in the explicitly hermitian form, $$K=\cos \pi N.
\label{kcos}$$ Hence, we have expressed the Klein operator in terms of the creation and annihilation operators, and, as a result, realized algebra (\[hei\]), (\[klein\]) as a deformation of the Heisenberg algebra.

To conclude this section, let us make the following remark. Using eqs. (\[a+a-n\]) and (\[kcos\]), one can write the relation $$N=a^{+}a^{-}+\frac{\nu}{2}(\cos\pi N -1).
\label{trans}$$ This relation is a transcendent equation for the number operator given as a function of the normal product $a^+a^-$. As a result, the defining relations of the $\nu$-deformed oscillator (\[hei\]), (\[klein\]) can be reduced to the form $$a^-a^+=g(a^+a^-),
\label{gener}$$ where $g(x)=x+f(x)$, and $f(x)=\nu\cos \pi N(x)$ is the function of $x=a^+a^-$. 