Danjun Zhao | Searcharxiv

arXiv2024

Roots and Logarithms of Multipliers

By now it is a well-known fact that if $f$ is a multiplier for the Drury-Arveson space $H^2_n$, and if there is a $c>0$ such that $|f(z)|\geq c$ for every $z\in B$, then the reciprocal function 1/f is also a multiplier for $H^2_n$. We show that for such an $f$ and for every $t\in \mathbb{R}$, $f^t$ is also a multiplier for $H^2_n$. We do so by deriving a differentiation formula for $R^m(f^th)$.Moreover, by this formula the same result holds for spaces $H_{m,s}$ of the Besov-Dirichlet type. The same technique also gives us the result that for a non-vanishing multiplier $f$ of $H^2_n$, $log f$ is a multiplier of $H^2_n$ if and only if log $f$ is bounded on $B$.

math.FA↗

arXiv2017

Derivations, local and 2-local derivations on some algebras of operators on Hilbert C*-modules

For a commutative C*-algebra $\mathcal A$ with unit $e$ and a Hilbert~$\mathcal A$-module $\mathcal M$, denote by End$_{\mathcal A}(\mathcal M)$ the algebra of all bounded $\mathcal A$-linear mappings on $\mathcal M$, and by End$^*_{\mathcal A}(\mathcal M)$ the algebra of all adjointable mappings on $\mathcal M$. We prove that if $\mathcal M$ is full, then each derivation on End$_{\mathcal A}(\mathcal M)$ is $\mathcal A$-linear, continuous, and inner, and each 2-local derivation on End$_{\mathcal A}(\mathcal M)$ or End$^{*}_{\mathcal A}(\mathcal M)$ is a derivation. If there exist $x_0$ in $\mathcal M$ and $f_0$ in $\mathcal M^{'}$, such that $f_0(x_0)=e$, where $\mathcal M^{'}$ denotes the set of all bounded $\mathcal A$-linear mappings from $\mathcal M$ to $\mathcal A$, then each $\mathcal A$-linear local derivation on End$_{\mathcal A}(\mathcal M)$ is a derivation.

math.OA↗