论文标题
$ n $ quasi的结构离开$ m $ iNvertible和相关的运营商类别
Structure of $n$-quasi left $m$-invertible and related classes of operators
论文作者
论文摘要
给定希尔伯特太空运营商$ t，s \ in \ b $，让$ \ triangle $和b（\ b）$ in B（\ b）$表示基本操作员$ \ triangle_ {t，s}（x）（x）=（l_tr_s-s-i） $δ_{t，s}（x）=（l_t-r_s）（x）= tx-xs $。令$ d = \三角形$或$δ$。假设$ t $上下班$ s^*$，并且选择$ x $作为积极的运算符$ s^{*n} s^n $用于某些正整数$ n $，本文利用了基本操作员的属性来研究$ n $ n $ quasi $ [m，d] $ $ d^m_ $ d^m_ d^m_ {t，s的结构运营商，其中包括$ n $ -quasi左$ m $ - iNvertible运营商，$ n $ -quasi $ m $ - iSometric operators，$ n $ quasi $ m $ m $ -m $ -selfdAdjoint Operators和$ n $ n $ -quasi $（m，c）$ symmetric operators（对于某些同种偶发$ c $ of $ c $ \ h $）。事实证明，$ s^n $是操作员的直接总和$ s_1^n =（s | _ {\ overline {s^n（\ h）}}）^n $满足$ d^m_ { $ t_1 = t | _ {\ overline {s^n（\ h）}} $，带有$ 0 $ operator;如果$ s $也使可逆，则$ s^n $类似于运算符$ b $，因此$ d^m_ {b^*，b}，b}（i）= 0 $。对于功率有限的$ s $和$ t $，使得$ st^* - t^*s = 0 $和$ \ triangle_ {t，s}（s^{*n} s^n）= 0 $，$ s $ is是宝丽来（即，光谱的隔离点是极点）。运营商$ t，s $满足$ d^m_ {t，s}（i）= 0 $的运营商$ t，s $的产品属性以及对某些通勤属性的扰动。
Given Hilbert space operators $T, S\in\B$, let $\triangle$ and $δ\in B(\B)$ denote the elementary operators $\triangle_{T,S}(X)=(L_TR_S-I)(X)=TXS-X$ and $δ_{T,S}(X)=(L_T-R_S)(X)=TX-XS$. Let $d=\triangle$ or $δ$. Assuming $T$ commutes with $S^*$, and choosing $X$ to be the positive operator $S^{*n}S^n$ for some positive integer $n$, this paper exploits properties of elementary operators to study the structure of $n$-quasi $[m,d]$-operators $d^m_{T,S}(X)=0$ to bring together, and improve upon, extant results for a number of classes of operators, amongst them $n$-quasi left $m$-invertible operators, $n$-quasi $m$-isometric operators, $n$-quasi $m$-selfadjoint operators and $n$-quasi $(m,C)$ symmetric operators (for some conjugation $C$ of $\H$). It is proved that $S^n$ is the perturbation by a nilpotent of the direct sum of an operator $S_1^n=(S|_{\overline{S^n(\H)}})^n$ satisfying $d^m_{T_1,S_1}(I_1)=0$, $T_1=T|_{\overline{S^n(\H)}}$, with the $0$ operator; if also $S$ is left invertible, then $S^n$ is similar to an operator $B$ such that $d^m_{B^*,B}(I)=0$. For power bounded $S$ and $T$ such that $ST^*-T^*S=0$ and $\triangle_{T,S}(S^{*n}S^n)=0$, $S$ is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators $T, S$ satisfying $d^m_{T,S}(I)=0$, given certain commutativity properties, transfers to operators satisfying $S^{*n}d^m_{T,S}(I)S^n=0$.
