# Journal of Operator Theory

Volume 83, Issue 2, Spring 2020 pp. 495-515.

Vacuum distribution, norm and spectral properties for sums of monotone position operators

**Authors**:
Vitonofrio Crismale (1), Yun Gang Lu (2)

**Author institution:**(1) Dipartimento di Matematica, Universit\`{a} degli studi di Bari, Via E. Orabona, 4, 70125 Bari, Italy

(2) Dipartimento di Matematica, Universit\`{a} degli studi di Bari, Via E. Orabona, 4, 70125 Bari, Italy

**Summary: **We investigate the spectrum for partial sums of $m$ position (or gaussian)
operators on monotone Fock space based on $\ell^2(\mathbb{N})$. In the
basic case of the first consecutive operators, we prove it coincides with the
support of the vacuum distribution. Thus, the right endpoint of the support
gives the norm. In the general case, we get that the last property for norm still holds. As any single position operator has the vacuum symmetric Bernoulli law, and the whole of them is a monotone independent family of random variables, the vacuum distribution for partial sums of $n$ operators can be seen as the monotone binomial with $n$ trials. It is a discrete measure supported on a finite set, and we exhibit recurrence formulas to compute its atoms and probability function as well. Moreover, lower and upper bounds for the right endpoints of the supports are given.

**DOI: **http://dx.doi.org/10.7900/jot.2018nov21.2215

**Keywords: **noncommutative probability, position operators, Gelfand spectrum, moment generating functions

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