### Abstract

We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are "close" if bounded functions of them are "close." We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. Also, these types of convergence strengthen the weak operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense.

Original language | English (US) |
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Pages (from-to) | 417-434 |

Number of pages | 18 |

Journal | Foundations of Physics |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1990 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Foundations of Physics*,

*20*(4), 417-434. https://doi.org/10.1007/BF00731710

**Convergence of observables on quantum logics.** / Tome, Wolfgang A.; Gudder, S.

Research output: Contribution to journal › Article

*Foundations of Physics*, vol. 20, no. 4, pp. 417-434. https://doi.org/10.1007/BF00731710

}

TY - JOUR

T1 - Convergence of observables on quantum logics

AU - Tome, Wolfgang A.

AU - Gudder, S.

PY - 1990/4

Y1 - 1990/4

N2 - We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are "close" if bounded functions of them are "close." We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. Also, these types of convergence strengthen the weak operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense.

AB - We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are "close" if bounded functions of them are "close." We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. Also, these types of convergence strengthen the weak operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense.

UR - http://www.scopus.com/inward/record.url?scp=34249954966&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249954966&partnerID=8YFLogxK

U2 - 10.1007/BF00731710

DO - 10.1007/BF00731710

M3 - Article

VL - 20

SP - 417

EP - 434

JO - Foundations of Physics

JF - Foundations of Physics

SN - 0015-9018

IS - 4

ER -