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Generalized Riesz Systems and Quasi Bases in Hilbert Space

Abstract

The purpose of this article is twofold. First of all, the notion of \(({\mathcal {D}}, {\mathcal {E}})\)-quasi basis is introduced for a pair \(({\mathcal {D}}, {\mathcal {E}})\) of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences \(\{ \varphi _n \}\) and \(\{ \psi _n \}\), such that \(\sum _{n=0}^\infty \left\langle {x}, {\varphi _n}\right\rangle \left\langle {\psi _n}, {y}\right\rangle =\left\langle {x}, {y}\right\rangle \) for all \(x \in {\mathcal {D}}\) and \(y \in {\mathcal {E}}\). Second, it is shown that if biorthogonal sequences \(\{ \varphi _n \}\) and \(\{ \psi _n \}\) form a \(({\mathcal {D}},{\mathcal {E}})\)-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant operators.

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Acknowledgements

This work was partially supported by the University of Palermo, by the Gruppo Nazionale per la Fisica Matematica (GNFM) and by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Correspondence to C. Trapani.

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Bagarello, F., Inoue, H. & Trapani, C. Generalized Riesz Systems and Quasi Bases in Hilbert Space. Mediterr. J. Math. 17, 41 (2020). https://doi.org/10.1007/s00009-019-1456-1

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Mathematics Subject Classification

  • 42B35
  • 47A07