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Full-Text Articles in Physical Sciences and Mathematics
On A Class Of Unitary Operators On The Bergman Space Of The Right Half Plane, Namita Das, Jitendra Kumar Behera
On A Class Of Unitary Operators On The Bergman Space Of The Right Half Plane, Namita Das, Jitendra Kumar Behera
Turkish Journal of Mathematics
In this paper, we introduce a class of unitary operators defined on the Bergman space $L_a^2(\mathbb{C}_+)$ of the right half plane $\mathbb{C}_+$ and study certain algebraic properties of these operators. Using these results, we then show that a bounded linear operator $S$ from $L_a^2(\mathbb{C}_+)$ into itself commutes with all the weighted composition operators $W_a, a \in \mathbb{D}$ if and only if $\widetilde{S}(w)=\langle Sb_{\overline{w}},b_{\overline{w}}\rangle, w \in \mathbb{C}_+ $ satisfies a certain averaging condition. Here for $a=c+id \in \mathbb{D}, f \in L_a^2(\mathbb{C}_+), W_af=(f \circ t_a) \frac{M^{\prime}}{M^{\prime} \circ t_a}, Ms=\frac{1-s}{1+s}, t_a(s)=\frac{-ids +(1-c)}{(1+c)s + id}$, and $b_{\overline{w}}(s)=\frac{1}{\sqrt{\pi}} \frac{1+w}{1+\overline{w}} \frac{2 \mbox {Re} w}{(s+w)^2}, w=M\overline {a}, …
When Is A Permutation Of The Set $\Z^N$ (Resp. $\Z_P^N$, $P$ Prime) An Automorphism Of The Group $\Z^N$ (Resp. $\Z_P^N$)?, Ben-Eben De Klerk, Johan H. Meyer
When Is A Permutation Of The Set $\Z^N$ (Resp. $\Z_P^N$, $P$ Prime) An Automorphism Of The Group $\Z^N$ (Resp. $\Z_P^N$)?, Ben-Eben De Klerk, Johan H. Meyer
Turkish Journal of Mathematics
For a given positive integer $n$, the structure, i.e. the number of cycles of various lengths, as well as possible chains, of the automorphisms of the groups $(\Z^n, +)$ and $(\Z_p^n,+)$, \ $p$ prime, is studied. In other words, necessary and sufficient conditions on a bijection $f : A \ra A$, where $ A $ is countably infinite (alternatively, of order $p^n$), are determined so that $A$ can be endowed with a binary operation $*$ such that $(A,*)$ is a group isomorphic to $(\Z^n,+)$ (alternatively, $(\Z_p^n,+)$) and such that $f\in \Aut(A)$.