arXiv subjects

Rongwei Yang

Publications and source records attributed to Rongwei Yang.

23 records · Page 2Linked to original sources

arXiv2018

Hilbert-Schmidtness of some finitely generated submodules in $H^2(\mathbb{D}^2)$

A closed subspace $\mathcal{M}$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions $z_1$ and $z_2$. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $\mathcal{M}$ containing $z_1 - φ(z_2)$ is Hilbert-Schmidt, where $φ$ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed.

math.FA↗

arXiv2017

Joint spectrum and infinite dihedral group

For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital Banach algebra ${\mathcal B}$, its {\em projective joint spectrum} $P(A)$ is the collection of $z\in {\mathbb C}^n$ such that the multiparameter pencil $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible. If ${\mathcal B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_1,\ A_2,\ ...,\ A_n$ with respect to a representation $ρ$, then $P(A)$ is an invariant of (weak) equivalence for $ρ$. This paper computes the joint spectrum of $(1,\ a,\ t)$ for the infinite dihedral group $D_{\infty}= $ with respect to the left regular representation $λ_D$, and gives an in-depth analysis on its properties. A formula for the Fuglede-Kadison determinant of the pencil $R(z)=1+z_1a+z_2t$ is obtained, and it is used to compute the first singular homology group of the joint resolvent set $P^c(R)$. The joint spectrum gives new insight into some earlier studies on groups of intermediate growth, through which the corresponding joint spectrum of $(1,\ a,\ t)$ with respect to the Koopman representation $ρ$ (constructed through a self-similar action of $D_{\infty}$ on a binary tree) can be computed. It turns out that the joint spectra with respect to the two representations coincide. Interestingly, this fact leads to a self-similar realization of the group $C^*$-algebra $C^*(D_{\infty})$. This self-similarity of $C^*(D_{\infty})$ is manifested by some dynamical properties of the joint spectrum.

math.GR↗

arXiv2013

Projective Spectrum and Cyclic Cohomolgy

For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital algebra ${\mathcal B}$ over $\mathbb{C}$, its {\em projective spectrum} $P(A)$ or $p(A)$ is the collection of $z\in \mathbb{C}^n$, or respectively $z\in \mathbb{P}^{n-1}$ such that the multi-parameter pencil $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible in ${\mathcal B}$. ${\mathcal B}$-valued $1$-form $A^{-1}(z)dA(z)$ contains much topological information about $P^c(A):=\mathbb{C}^n\setminus P(A)$. In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of ${\mathcal B}$ does a similar job. In fact, a Chen-Weil type map $κ$ from the cyclic cohomology of ${\mathcal B}$ to the de Rham cohomology $H^*_d(P^c(A),\ \mathbb{C})$ is established. As an example, we prove a closed high-order form of the classical Jacobi's formula.

math.FA↗

arXiv2008

Projective spectrum in Banach algebras

For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${\mathcal B}$, its {\em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]\in \pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ is not invertible in ${\mathcal B}$. The pre-image of $p(A)$ in ${\cc}^{n+1}$ is denoted by $P(A)$. When ${\mathcal B}$ is the $k\times k$ matrix algebra $M_k(\cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When ${\mathcal B}$ is reflexive or is a $C^*$-algebra, the {\em projective resolvent set} $P^c(A):=\cc^{n+1}\setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${\mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ on $P^c(A)$. As a consequence, we show that if ${\mathcal B}$ is a $C^*$-algebra with a trace $ϕ$, then $ϕ(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), \cc)$.

math.FA↗

arXiv2007

$N_{\p}$-type quotient modules on the torus

Structure of the quotient modules in $\hh$ is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-call $N_{\p}$-type quotient modules, namely, quotient modules of the form $\hh\ominus [z-\p]$, where $\p (w)$ is a function in the classical Hardy space $H^2(\G)$ and $[z-\p]$ is the submodule generated by $z-\p (w)$. This type of quotient modules serve as good examples in many studies. A notable feature of the $N_{\p}$-type quotient module is its close connections with some classical single variable operator theories.

math.FA↗