arXiv subjects
Publications and source records attributed to Mengmeng Zhou.
Several upper bounds as well as one lower bound for the operator norm of the generalized Hilbert operator $\mathcal{H}_b$ acting on weighted Bergman spaces $A_{\alpha}^p$ are established. Moreover, under some mild assumptions, we obtain the exact norm of $\mathcal{H}_b$ on $ A_{\alpha}^p$.
We study the generalized Hilbert operator \[ \mathcal{H}_b f(z)=\int_0^1 f(t)\,\frac{(1-t)^b}{(1-tz)^{b+1}}\,dt, \qquad b>0, \] acting on the Hardy spaces $H^p$ for $1\leq p\leq \infty$. We establish the precise operator norm \[ \|\mathcal{H}_b\|_{H^p\to H^p}=B\!\left(\frac1p,b+1-\frac1p\right) \] for every $1 0$, in contrast with the classical Hilbert operator, and we obtain the sharp restricted norm estimate \[ \|\mathcal{H}_b\|_{H^1_0\to H^1}=B(1,b). \] We also determine the exact norm \[ \|\mathcal{H}_b\|_{H^\infty\to \mathcal B}=\frac{1}{b+1}+2. \]
Let $A_\alpha^p$ be the weighted Bergman space on the unit disk, where $\alpha>-1$. For $f(z)=\sum_{k=0}^{\infty}a_k z^k\in A_\alpha^p$, consider the Hilbert matrix operator $\mathcal{H}f(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\frac{a_k}{n+k+1}\right)z^n =\int_0^1\frac{f(t)}{1-tz}\,dt$. For even exponents $p=2m$, we prove that $\|\mathcal{H}\|_{A_\alpha^{2m}\to A_\alpha^{2m}}=B(a,1-a)$, where $a=(\alpha+2)/(2m)$, whenever $0 B(a_0,1-a_0)$. The counterexample is based on the fixed function $f_0(z)=(1-z^2)^{-4/5}=\sum_{k=0}^{\infty}\frac{(4/5)_k}{k!}z^{2k}$. A rigorous interval estimate at $p=1100000$, together with monotonicity in $p$, yields the result on the entire half-line. In particular, the formula fails for every even exponent $p=2m$ with $m\geq 550000$.
This paper studies the concept of stable perturbation $B\in\mathbb{C}^{n\times n}$ for the core-EP inverse of a matrix $A\in\mathbb{C}^{n\times n}$ with index $k$. For a given stable perturbation $B$ of $A$, explicit expressions of its core-EP inverse $B^{\scriptsize\textcircled{\tiny $\dagger$}}$ and its projection at zero $B^π$ are presented. Then, the perturbation bounds of $\parallel B^{\scriptsize\textcircled{\tiny $\dagger$}}-A^{\scriptsize\textcircled{\tiny $\dagger$}}\parallel/\parallel A^{\scriptsize\textcircled{\tiny $\dagger$}}\parallel$ and $\parallel B^π-A^π\parallel$ are given provided that $B$ is a stable perturbation of $A$. In addition, we investigate the concept of acute perturbation of $A$. We give a perturbation analysis with respect to core-EP inverses. We provide a condition under which the acute perturbation coincides with the stable perturbation for core-EP inverses.
In a unitary ring with involution, we prove that each element has at most one weak group inverse if and only if each idempotent element has a unique weak group inverse. Furthermore, we define the $m$-weak group inverse and show some properties of $m$-weak group inverse.
Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $φ: X \rightarrow X$ is a morphism with kernel $κ: K \rightarrow X$ in $\mathscr{C}$, then $φ$ is core invertible if and only if $φ$ has a cokernel $λ: X \rightarrow L$ and both $κλ$ and $φ^{\ast}φ^3+κ^{\ast}κ$ are invertible. In this case, we give the representation of the core inverse of $φ$. We also give the corresponding result about dual core inverse.
In this paper, we present three limit representations of the core-EP inverse. The first approach is based on the full-rank decomposition of a given matrix. The second and third approaches, which depend on the explicit expression of the core-EP inverse, are established. The corresponding limit representations of the dual core-EP inverse are also given. In particular, limit representations of the core and dual core inverse are derived