arXiv subjects

Hyuga Ito

Publications and source records attributed to Hyuga Ito.

5 recordsLinked to original sources

arXiv2026

NC functions over the nc Grassmannian

We explain how to formulate Voiculescu's non-commutative Riemann sphere framework for fully matricial functions \cite{v10} within the theory of nc functions developed by Vinnikov and Kaliuzhnyi-Verbovetskyi \cite{kvv14}. We then extend this framework from the Riemann sphere to Grassmannians (and flag manifolds). Moreover, as an example of nc functions in this setting, we introduce a generalization of Voiculescu's non-commutative resolvent on the Riemann sphere, study a corresponding generalization of the resolvent equation, and discuss aspects of the spectral analysis of unbounded operators in Voiculescu's framework \cite{v10}.

math.FA↗

arXiv2024

$B$-valued semi-circular system and the free Poincar\'{e} inequality

We characterize $B$-valued semi-circular system in terms of $B$-valued free probabilistic analogue of Poincar\'{e} inequality. This is a $B$-valued generalization of Biane's theorem \cite[Theorem 5.1]{b03}. Moreover, we prove that Voiculescu's conjecture on $B$-valued free Poincar\'{e} inequality in \cite{aim06} is not in the affirmative as it is.

math.OA↗

arXiv2023

A note on free divergence-free vector fields

We exhibit an orthonormal basis of cyclic gradients and a (non-orthogonal) basis of the homogeneous free divergence-free vector field on the full Fock space and determine the dimension of Voiculescu's free divergence-free vector field of degree k or less. Moreover, we also give a concrete formula for the orthogonal projection onto the space of cyclic gradients as well as the free Leray projection.

math.OA↗

arXiv2023

Differential calculus for fully matricial functions I

We will introduce a cyclic derivative for fully (stably) matricial functions and study its basic properties. In particular, we will show the Poincaré lemma for stably matricial functions of certain classes. We will also position Voiculescu's framework of fully matricial functions in the context of nc functions due to Kaliuzhnyi-Verbovetskyi and Vinnikov in order to clarify the relation between the present work and previous related works.

math.OA↗

arXiv2023

An operator-coefficients free Poincaré inequality

We prove the following operator-coefficients free Poincaré inequality: $$|f(X)-E[f(X)]|_{2}\leq2|X|_{2}\|\widehat{\partial}_{X:B}[f(X)]\|_π, \quad f(X) \in \mathrm{dom}(\widehat{\partial}_{X:B}),$$ where $\|\cdot\|_π$ is the projective tensor norm.

math.OA↗