In this paper we introduce a general theory of regular biorthogonal sequences and its physical applications. Biorthogonal sequences $\{ \phi_{ n} \}$ and $\{ \psi_{ n} \}$ in a Hilbert space ${ \cal H}$ are said to be regular if $Span\; \{ \phi_{ n} \}$ and $Span\; \{ \psi_{ n} \}$ are dense in ${ \cal H}$. The first purpose is to show that there exists a non-singular positive self-adjoint operator $T_{ \mbox{ $f$}}$ in ${ \cal H}$ defined by an ONB $\mbox{ $f$} \equiv \{ f_{ n} \}$ in ${ \cal H}$ such that $\phi_{ n}=T_{ \mbox{ $f$}} f_{ n}$ and $\psi_{ n}= T_{ \mbox{ $f$}}^{ -1} f_{ n}$, $n=0,1, \cdots$, and such an ONB $\mbox{ $f$}$ is unique. The second purpose is to define and study the lowering operators $A_{ \mbox{ $f$}}$ and $B_{ \mbox{ $f$}}^{ \dagger}$, the raising operators $B_{ \mbox{ $f$}}$ and $A_{ \mbox{ $f$}}^{ \dagger}$, the number operators $N_{ \mbox{ $f$}}$ and $N_{ \mbox{ $f$}}^{ \dagger}$ determined by the non-singular positive self-adjoint operator $T_{ \mbox{ $f$}}$. These operators connect with ${ \it quasi}$-${ \it hermitian \; quantum \; mechanics}$ and its relatives. This paper clarifies and simplifies the mathematical structure of this framework minimized the required assumptions.
Categories: math-ph math.MP