Journal of the Iranian Mathematical Society
https://jims.ims.ir/
Journal of the Iranian Mathematical Societyendaily1Fri, 01 Mar 2024 00:00:00 +0330Fri, 01 Mar 2024 00:00:00 +0330Conciseness on normal subgroups and new concise words from outer commutator words
https://jims.ims.ir/article_191502.html
Let $w=w(x_1,\ldots,x_r)$ be an outer commutator word&lrm;.&nbsp;&lrm;We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables&lrm;.&nbsp;&lrm;This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$&lrm;, &lrm;where the $n_i$ are non-zero integers&lrm;.&nbsp;&lrm;Our approach is via the study of values of $w$ on normal subgroups&lrm;, &lrm;and in this setting, &lrm;we obtain the following result&lrm;: &lrm;if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all&lrm; &lrm;values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite, &lrm;then also the subgroup generated by these values&lrm;, &lrm;i.e. $w(N_1,\ldots,N_r)$&lrm;, &lrm;is finite&lrm;.Characterization of the structured pseudospectrum in non-Archimedean Banach spaces
https://jims.ims.ir/article_194146.html
In this paper, we demonstrate some results on the pseudospectrum of linear operator pencils on non-Archimedean Banach spaces. In particular, we give a relationship between the Fredholm spectrum of a bounded operator pencil $(A,B)$ and the Fredholm spectrum of the pencil $(A^{-1},B^{-1}).$ Also, we establish a characterization of the essential spectrum of operator pencils on non-Archimedean Banach spaces. Furthermore, we introduce and study the structured pseudospectrum of linear operators on non-Archimedean Banach spaces. We prove that the structured pseudospectra associated with various $\varepsilon$ are nested sets and the intersection of all the structured pseudospectra is the spectrum. We establish a characterization of the structured pseudospectrum of bounded linear operators on non-Archimedean Banach spaces. Finally, we characterize the structured essential pseudospectrum of bounded linear operator pencils on non-Archimedean Banach spaces and we give an illustrative example.