of the creation-annihilation operators themselves [@pld; @mac], and in this case algebra (\[hei\]), (\[klein\]) gives a specific ‘$\nu$-deformed’ bosonic oscillator. We shall understand algebra (\[hei\]), (\[klein\]) in this latter sense, i.e. as a deformed Heisenberg algebra.

Let us introduce the Fock-type vacuum, $$a^{-}|0>=0,\quad
<0|0>=1,
\quad K|0>=\kappa|0>,$$ where $\kappa=+1$ or $-1$. Without loss of generality, we put here $\kappa=+1$. Then we get the action of the operator $a^{+}a^{-}$ on the states $(a^{+})^{n}|0>$, $n=0,1,\ldots$, $$a^{+}a^{-}(a^{+})^{n}|0>=[n]_{\nu}(a^{+})^{n}|0>,
\label{a+a-}$$ where $$[n]_{\nu}=n+\frac{\nu}{2}
\left(1+(-1)^{n+1}\right).$$ From here we conclude that in the case when $$\nu >-1,
\label{restr}$$ the space of unitary representation of algebra (\[hei\]), (\[klein\]) is given by the complete set of the normalized vectors $$|n>=\frac{1}{\sqrt{[n]_{\nu}!}}(a^{+})^{n}|0>,\quad
<n|n'>=\delta_{nn'},$$ where $$[n]_{\nu}!=\prod_{k=1}^{n}[k]_{\nu}.$$ In correspondence with eqs. (\[klein\]), the Klein operator $K$ separates the complete set of states $|n>$ into even and odd subspaces: $$K|n>=(-1)^{n}|n>,
\label{z2}$$ and, so, introduces $Z_2$-grading structure on the Fock space of the deformed ($\nu\neq 0$) or ordinary ($\nu=0$) bosonic oscillator. Let us introduce the operators $\Pi_{+}$ and $\Pi_{-}$, $$\Pi_{\pm}=\frac{1}{2}(1\pm K),
\label{proj}$$ which satisfy the equalities $\Pi_{\pm}^{2}=\Pi_{\pm},$ $\Pi_{+}\Pi_{-}=0,$ $\Pi_{+}+\Pi_{-}=1$, and are the projector operators on the even and odd subspaces of the total Fock space, respectively. Then, from eqs. (\[a+a-\]) and (\[z2\]) we get the equality: $$a^{+}a^{-}=N+\nu\Pi_{-},
