Towards Generalized Riesz Systems Theory

Kamuda, Alan; Kużel, Sergiusz

doi:10.1007/s11785-020-00987-3

Cited by 5 publications

(3 citation statements)

References 31 publications

(58 reference statements)

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“…φ is actually an orthonormal basis). A particular case of this question has been treated, in the discrete case and with a different approach, in [25].…”

Section: Introductionmentioning

confidence: 99%

Lower Semi-frames, Frames, and Metric Operators

2020

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This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

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“…φ is actually an orthonormal basis). A particular case of this question has been treated, in the discrete case and with a different approach, in [25].…”

Section: Introductionmentioning

confidence: 99%

Lower Semi-frames, Frames, and Metric Operators

2020

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“…In particular we mention generalized Riesz systems introduced by one of us (H.I) and analyzed in a series of papers [7][8][9][10][11][12][13][14]). For other studies on extensions of Riesz bases or on generalizations to different environments (Krein spaces, Rigged Hilbert spaces) we refer to [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Gibbs States, Algebraic Dynamics and Generalized Riesz Systems

Багарелло

Inoué

Trapani

2020

Complex Anal. Oper. Theory

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In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita–Takesaki theory in our context.

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“…In particular we mention generalized Riesz systems introduced by one of us (H.I) and analyzed in a series of papers [13,14,15,19,20,21,22,23]). For other studies on extensions of Riesz bases or on generalizations to different environments (Krein spaces, Rigged Hilbert spaces) we refer to [11,12,17].…”

Section: Introductionmentioning

confidence: 99%