arXiv subjects

M. N. Reshmi

Publications and source records attributed to M. N. Reshmi.

3 recordsLinked to original sources

arXiv2026

Connecting $H^\infty$-functional calculus and isometric dilations for commuting families of Ritt$_E$ operators

Let $(T_1,\ldots,T_d)$ be a commuting $d$-tuple of Ritt$_E$ operators on some UMD Banach space $X$. We show that $(T_1,\ldots,T_d)$ admits a bounded $H^\infty$-functional calculus if and only if $T_k$ is an $R$-Ritt$_E$ operator for every $k=1,\ldots,d$, and the $d$-tuple $(T_1,\ldots,T_d)$ admits an isometric dilation $(U_1,\ldots,U_d)$ on some UMD Banach space $Y$ such that $(U_1,\ldots,U_d)$ is polynomially bounded. In the case where $X$ further possesses property $(α)$, we establish other characterizations of the $H^\infty$-functional calculus property for $(T_1,\ldots,T_d)$ in terms of isometric dilations.

math.FA↗

arXiv2025

Commuting families of polygonal type operators on Hilbert space

Let $T\colon H\to H$ be a bounded operator on Hilbert space. We say that $T$ has a polygonal type if there exists an open convex polygon $Δ\subset {\mathbb D}$, with $\overlineΔ\cap{\mathbb T}\neq\emptyset$, such that the spectrum $σ(T)$ is included in $\overlineΔ$ and the resolvent $R(z,T)$ satisfies an estimate $\Vert R(z,T)\Vert \lesssim \max\{\vert z-ξ\vert^{-1}\, :\, ξ\in \overlineΔ\cap{\mathbb T}\}$ for $z\in\overline{\mathbb D}^c$. The class of polygonal type operators (which goes back to De Laubenfels and Franks-McIntosh) contains the class of Ritt operators. Let $T_1,\ldots,T_d$ be commuting operators on $H$, with $d\geq 3$. We prove functional calculus properties of the $d$-tuple $(T_1,\ldots,T_d)$ under various assumptions involving poygonal type. The main ones are the following. (1) If the $T_k$ are contractions for all $k=1,\ldots,d$ and if $T_1,\ldots,T_{d-2}$ have a polygonal type, then $(T_1,\ldots,T_d)$ satisfies a generalized von Neumann inequality $\Vert ϕ(T_1,\ldots,T_d)\Vert \leq C\Vertϕ\Vert_{\infty,{\mathbb D}^d}$ for polynomials $ϕ$ in $d$ variables; (2) If $T_k$ is polynomially bounded with a polygonal type for all $k=1,\ldots,d$, then there exists an invertible operator $S\colon H\to H$ such that $\Vert S^{-1}T_kS\Vert \leq 1$ for all $k=1,\ldots,d$.

math.FA↗

arXiv2022

Factorization of Characteristic Functions of Iterated Liftings

We obtain a factorization of the characteristic function of a contractive two-step iterated lifting in terms of the characteristic functions of constituent liftings of the iterated lifting and the Julia-Halmos matrix. We also give an expression for the characteristic function of the minimal part of a contractive two-step iterated lifting as a restriction of the product of the characteristic functions of constituent liftings of the iterated lifting.

math.FA↗