On the dimension of the algebra generated by two positive semi-commuting matrices

详细信息    查看全文

• 作者：Marko Kandić ^{marko.kandic@fmf.uni-lj.si" class="auth_mail" title="E-mail the corresponding author} ; Klemen &Scaron ; ivic ; ^{klemen.sivic@fmf.uni-lj.si" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A27 ; 15B48 ; 47B47
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：512
• 期：Complete
• 页码：136-161
• 全文大小：456 K

文摘

Gerstenhaber's theorem states that the dimension of the unital algebra generated by two commuting si1" class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304268&_mathId=si1.gif&_user=111111111&_pii=S0024379516304268&_rdoc=1&_issn=00243795&md5=e0d766e2f74d1474a8833bb0a1d95308" title="Click to view the MathML source">n×nclass="mathContainer hidden">class="mathCode"> matrices is at most n  . We study the analog of this question for positive matrices with a positive commutator. We show that the dimension of the unital algebra generated by the matrices is at most si119" class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304268&_mathId=si119.gif&_user=111111111&_pii=S0024379516304268&_rdoc=1&_issn=00243795&md5=cbce7ada8145fdfec3789baabd4e572a">class="imgLazyJSB inlineImage" height="22" width="46" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516304268-si119.gif">class="mathContainer hidden">class="mathCode"> and that this bound can be attained. We also consider the corresponding question if one of the matrices is a permutation or a companion matrix or both of them are idempotents. In these cases, the upper bound for the dimension can be reduced significantly. In particular, the unital algebra generated by two semi-commuting positive idempotent matrices is at most 9-dimensional. This upper bound can be attained.