The first purpose of this paper is to show a method of constructing a regular biorthogonal pair based on the commutation rule: $ab-ba=I$ for a pair of operators $a$ and $b$ acting on a Hilbert space ${ \cal H}$ with inner product $( \cdot | \cdot )$. Here, sequences $\{ \phi_{ n} \}$ and $\{ \psi_{ n} \}$ in a Hilbert space ${ \cal H}$ are biorthogonal if $( \phi_{ n} | \psi_{ m})= \delta_{ nm}$, $n,m=0,1, \cdots$, and they are regular if both $D_{ \phi} \equiv Span \{ \phi_{ n} \}$ and $D_{ \psi} \equiv Span \{ \psi_{ n} \}$ are dense in ${ \cal H}$. Indeed, the assumption to construct the regular biorthogonal pair coincide with the definition of pseudo-bosons as originally given in Ref \cite{ bagarello10}. Furthermore, we study the connections between the pseudo-bosonic operators $a, \; b, \; a^{ \dagger}, \; b^{ \dagger}$ and the pseudo-bosonic operators defined by a regular biorthogonal pair $(\{ \phi_{ n} \}$, $\{ \psi_{ n} \} )$ and an ONB $\mbox{ $e$}$ of ${ \cal H}$ in appeared Ref \cite{ hiroshi1}. The second purpose is to define and study the notion of ${ \cal D}$-pseudo bosons in Ref \cite{ bagarello13, bagarello2013} and give a method of constructing ${ \cal D}$-pseudo bosons on some steps. Then it is shown that for any ONB $\mbox{ $e$}= \{ e_{ n} \}$ in ${ \cal H}$ and any operators $T$ and $T^{ -1}$ in ${ \cal L}^{ \dagger} ( { \cal D})$, we may construct operators $A$ and $B$ satisfying ${ \cal D}$-pseudo bosons, where ${ \cal D}$ is a dense subspace in a Hilbert space ${ \cal H}$ and ${ \cal L}^{ \dagger} ( { \cal D})$ the set of all linear operators $T$ from ${ \cal D}$ to ${ \cal D}$ such that $T^{ \ast} { \cal D} \subset { \cal D}$, where $T^{ \ast}$ is the adjoint of $T$. Finally, we give some physical examples of ${ \cal D}$-pseudo bosons based on standard bosons by the method of constructing ${ \cal D}$-pseudo bosons stated above.
Categories: math-ph math.MP