### Abstract

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
with inner product (⋅| ⋅ ). Here, sequences {ϕn} and {ψn} in a Hilbert space
are biorthogonal if (ϕn|ψm) = δnm, n, m = 0, 1, …, and they are regular if both D
ϕ ≡ Span{ϕn} and D
ψ ≡ Span{ψn} are dense in . Indeed, the assumptions to construct the regular biorthogonal pair coincide with the definition of pseudo-bosons as originally given in F. Bagarello [“Pseudobosons, Riesz bases, and coherent states,” J. Math. Phys. 51, 023531 (2010)]. Furthermore, we study the connections between the pseudo-bosonic operators a, b, a
^{†}, b
^{†} and the pseudo-bosonic operators defined by a regular biorthogonal pair ({ϕn}, {ψn}) and an ONB e of in H. Inoue [“General theory of regular biorthogonal pairs and its physical applications,” e-print arXiv:math-ph/1604.01967]. The second purpose is to define and study the notion of -pseudo-bosons in F. Bagarello [“More mathematics for pseudo-bosons,” J. Math. Phys. 54, 063512 (2013)] and F. Bagarello [“From self-adjoint to non self-adjoint harmonic oscillators: Physical consequences and mathematical pitfalls,” Phys. Rev. A 88, 032120 (2013)] and give a method of constructing -pseudo-bosons on some steps. Then it is shown that for any ONB e = {en} in and any operators T and T
^{−1} in , we may construct operators A and B satisfying -pseudo bosons, where is a dense subspace in a Hilbert space
and the set of all linear operators T from to such that , where T
^{*} is the adjoint of T. Finally, we give some physical examples of -pseudo-bosons based on standard bosons by the method of constructing -pseudo-bosons stated above.

Published by AIP Publishing.

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