# Journal of Operator Theory

Volume 85, Issue 1, Winter 2021 pp. 21-43.

Unimodality for free multiplicative convolution with free normal distributions on the unit circle

**Authors**:
Takahiro Hasebe (1), Yuki Ueda (2)

**Author institution:** (1) Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

(2) Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

**Summary: **We study unimodality for free multiplicative convolution with free normal distributions $\{\lambda_t\}_{t>0}$ on the unit circle. We give four results on unimodality for $\mu\boxtimes\lambda_t$: (1) if $\mu$ is a symmetric unimodal distribution on the unit circle then so is $\mu\boxtimes \lambda_t$ at any time $t>0$; (2) if $\mu$ is a symmetric distribution on $\mathbb{T}$ supported on $\{\mathrm e^{\mathrm i\theta}: \theta \in [-\varphi,\varphi]\}$ for some $\varphi \in (0,\frac{\pi}{2})$, then $\mu \boxtimes \lambda_t$ is unimodal for sufficiently large $t>0$; (3) $\mathbf b \boxtimes \lambda_t$ is not unimodal at any time $t>0$, where $\mathbf b$ is the equally weighted Bernoulli distribution on $\{1,-1\}$; (4) $\lambda_t$ is not freely strongly unimodal for sufficiently small $t>0$. Moreover, we study unimodality for classical multiplicative convolution, which is useful in proving the above four results.

**DOI: **http://dx.doi.org/10.7900/jot.2019mar23.2264

**Keywords: **classical/free multiplicative convolution, Poisson kernel, free normal distribution on the unit circle, unimodality, classical/free strong unimodality

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