Extremality of numerical radii of matrix products

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• 作者：Hwa-Long Gau^a ; ¹ ; ^{hlgau@math.ncu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Pei Yuan Wu^b ; ^{pywu@math.nctu.edu.tw" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A60 ; 15A27
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：501
• 期：Complete
• 页码：17-36
• 全文大小：420 K

文摘

For two n-by-n matrices A and B  , it was known before that their numerical radii satisfy the inequality lsi1" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si1.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=61a01a813ddd5dd90e6d05f03fe0bb44" title="Click to view the MathML source">w(AB)≤4w(A)w(B)lass="mathContainer hidden">lass="mathCode">low="scroll">wlse">(ABlse">)≤4wlse">(Alse">)wlse">(Blse">), and the equality is attained by the 2-by-2 matrices lsi2" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si2.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=d20ee60949ff7061a5e52ac978e5c60c">lass="imgLazyJSB inlineImage" height="21" width="66" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300076-si2.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">A=[le>lumnalign="center">0lumnalign="center">1lumnalign="center">0lumnalign="center">0le>] and lsi3" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si3.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=9ebdfe37d20ebdd73bdd5388ec7aac79">lass="imgLazyJSB inlineImage" height="21" width="67" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300076-si3.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">B=[le>lumnalign="center">0lumnalign="center">0lumnalign="center">1lumnalign="center">0le>]. Moreover, the constant “4” here can be reduced to “2” if A and B   commute, and the corresponding equality is attained by lsi4" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si4.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=55ea59825a8be093ee3c3cb00b1714a4">lass="imgLazyJSB inlineImage" height="21" width="103" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300076-si4.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">A=I2⊗[le>lumnalign="center">0lumnalign="center">1lumnalign="center">0lumnalign="center">0le>] and lsi5" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si5.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=3f1fa01217b032ae75607c3ae63a1132">lass="imgLazyJSB inlineImage" height="21" width="106" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300076-si5.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">B=[le>lumnalign="center">0lumnalign="center">1lumnalign="center">0lumnalign="center">0le>]⊗I2. In this paper, we give a complete characterization of A and B   for which the equality holds in each case. More precisely, it is shown that lsi57" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si57.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=bde2959537683c3517043bf45f3c0c1f" title="Click to view the MathML source">w(AB)=4w(A)w(B)lass="mathContainer hidden">lass="mathCode">low="scroll">wlse">(ABlse">)=4wlse">(Alse">)wlse">(Blse">) (resp., lsi159" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si159.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=4ecdc2a702fe23c512872e6aad6d4c46" title="Click to view the MathML source">w(AB)=2w(A)w(B)lass="mathContainer hidden">lass="mathCode">low="scroll">wlse">(ABlse">)=2wlse">(Alse">)wlse">(Blse">) for commuting A and B) if and only if either A or B is the zero matrix, or A and B   are simultaneously unitarily similar to matrices of the form lsi26" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si26.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=a7093c9bf96862e757ad398658340dad">lass="imgLazyJSB inlineImage" height="21" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300076-si26.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">[le>lumnalign="center">0lumnalign="center">alumnalign="center">0lumnalign="center">0le>]⊕A′ and lsi27" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si27.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=eb3d18a08cef7698a303c3b92b58e119">lass="imgLazyJSB inlineImage" height="21" width="72" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300076-si27.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">[le>lumnalign="center">0lumnalign="center">0lumnalign="center">blumnalign="center">0le>]⊕B′ (resp., with lsi11" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si11.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=1e229d0f94852204a933e39eec5bc835" title="Click to view the MathML source">w(A^′)≤|a|/2lass="mathContainer hidden">lass="mathCode">low="scroll">wlse">(A′lse">)≤lse">|alse">|lse">/2 and lsi12" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si12.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=164ad8b64e6d682a98ac701b0d6e19d5" title="Click to view the MathML source">w(B^′)≤|b|/2lass="mathContainer hidden">lass="mathCode">low="scroll">wlse">(B′lse">)≤lse">|blse">|lse">/2. An analogous characterization for the extremal equality for tensor products is also proven. For doubly commuting matrices, we use their unitary similarity model to obtain the corresponding result. For commuting 2-by-2 matrices A and B  , we show that lsi13" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si13.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=f97c4b18274f69cb2bc0035322ca86a1" title="Click to view the MathML source">w(AB)=w(A)w(B)lass="mathContainer hidden">lass="mathCode">low="scroll">wlse">(ABlse">)=wlse">(Alse">)wlse">(Blse">) if and only if either A or B is a scalar matrix, or A and B   are simultaneously unitarily similar to lsi14" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si14.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=14ace1731c9857477311226c5cbe8450">lass="imgLazyJSB inlineImage" height="29" width="46" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300076-si14.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">[le>lumnalign="center">a1lumnalign="center">0lumnalign="center">0lumnalign="center">a2le>] and lsi15" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si15.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=27e83bfbf0dc5c7b801a2b4d06375b28">lass="imgLazyJSB inlineImage" height="29" width="44" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300076-si15.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">[le>lumnalign="center">b1lumnalign="center">0lumnalign="center">0lumnalign="center">b2le>] with lsi16" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si16.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=c2023d12b941a19a61824e6063cc325f" title="Click to view the MathML source">|a₁|≥|a₂|lass="mathContainer hidden">lass="mathCode">low="scroll">lse">|a1lse">|≥lse">|a2lse">| and lsi17" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300076&_mathId=si17.gif&_user=111111111&_pii=S0024379516300076&_rdoc=1&_issn=00243795&md5=e89e65508fd5889c24ebe41a8ad6aab3" title="Click to view the MathML source">|b₁|≥|b₂|lass="mathContainer hidden">lass="mathCode">low="scroll">lse">|b1lse">|≥lse">|b2lse">|.