{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20056"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20056","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Analysis of an inequality concerning perturbation of self-adjoint operators","abstract":"In 1983, Bhatia, Davis and McIntosh proved that if A and B are self-adjoint with dist($\\sigma(A),\\sigma(B)) \\geq \\delta$ then there is some $c\\sb{sa} < 2$ (independent of A and B) such that $c\\sb{sa}\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert \\geq \\delta\\vert\\vert\\vert Q\\vert\\vert\\vert$ for any Q and any unitary invariant norm $\\vert\\vert\\vert \\cdot\\vert\\vert\\vert$. Sz.-Nagy subsequently noted that $c\\sb{sa} \\leq \\pi$/2. It has not been known, however, if $\\pi/2$ is sharp. In this dissertation we analyze $\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert$ relative to $\\vert\\vert\\vert Q\\vert\\vert\\vert$ in certain special cases in order to sharpen this inequality. We define $c\\sb{n}$ as the smallest constant such that $c\\sb{n}\\Vert AQ - QB\\Vert \\geq \\delta\\Vert Q\\Vert$ when A, Q and B are $n \\times n$ matrices and $\\Vert \\cdot\\Vert$ is the usual norm. We then prove $c\\sb2$ = $\\sqrt{3/2}\\approx$ 1.22474, $c\\sb3$ = (8 + 5$\\sqrt{10}$)/18 $\\approx$ 1.32285, and $\\lim\\limits\\sb{n \\to \\infty}$ $c\\sb{n}$ = $\\pi$/2 $\\approx$ 1.57080. In particular, this proves $\\pi$/2 is sharp in this operator inequality.","abstract_html":"In 1983, Bhatia, Davis and McIntosh proved that if A and B are self-adjoint with dist(<span class=\"etd-inline-math\">&sigma;(A),&sigma;(B)) \\geq &delta;</span> then there is some $c\\sb{sa} &lt; 2$ (independent of A and B) such that <span class=\"etd-inline-math\">c\\sb{sa}\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert \\geq &delta;\\vert\\vert\\vert Q\\vert\\vert\\vert</span> for any Q and any unitary invariant norm $\\vert\\vert\\vert \\cdot\\vert\\vert\\vert$. Sz.-Nagy subsequently noted that <span class=\"etd-inline-math\">c\\sb{sa} \\leq &pi;</span>/2. It has not been known, however, if <span class=\"etd-inline-math\">&pi;/2</span> is sharp. In this dissertation we analyze $\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert$ relative to $\\vert\\vert\\vert Q\\vert\\vert\\vert$ in certain special cases in order to sharpen this inequality. We define $c\\sb{n}$ as the smallest constant such that <span class=\"etd-inline-math\">c\\sb{n}\\Vert AQ - QB\\Vert \\geq &delta;\\Vert Q\\Vert</span> when A, Q and B are $n \\times n$ matrices and $\\Vert \\cdot\\Vert$ is the usual norm. We then prove $c\\sb2$ = $\\sqrt{3/2}\\approx$ 1.22474, $c\\sb3$ = (8 + 5$\\sqrt{10}$)/18 $\\approx$ 1.32285, and $\\lim\\limits\\sb{n \\to \\infty}$ $c\\sb{n}$ = <span class=\"etd-inline-math\">&pi;</span>/2 $\\approx$ 1.57080. In particular, this proves <span class=\"etd-inline-math\">&pi;</span>/2 is sharp in this operator inequality.","abstract_has_math":true,"creators":["McEachin, Raymond Vincent, Jr"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:27:26Z","date_published":"2011-05-07T12:27:26Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 McEachin, Raymond Vincent, Jr"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114341","(UMI)AAI9114341"],"render_values":[{"text":"AAI9114341","href":null,"code":true},{"text":"(UMI)AAI9114341","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20056","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["McEachin, Raymond Vincent, Jr"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:27:26Z","10000-01-01","1990"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 McEachin, Raymond Vincent, Jr"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114341","(UMI)AAI9114341","http://hdl.handle.net/2142/20056"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In 1983, Bhatia, Davis and McIntosh proved that if A and B are self-adjoint with dist($\\sigma(A),\\sigma(B)) \\geq \\delta$ then there is some $c\\sb{sa} < 2$ (independent of A and B) such that $c\\sb{sa}\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert \\geq \\delta\\vert\\vert\\vert Q\\vert\\vert\\vert$ for any Q and any unitary invariant norm $\\vert\\vert\\vert \\cdot\\vert\\vert\\vert$. Sz.-Nagy subsequently noted that $c\\sb{sa} \\leq \\pi$/2. It has not been known, however, if $\\pi/2$ is sharp. In this dissertation we analyze $\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert$ relative to $\\vert\\vert\\vert Q\\vert\\vert\\vert$ in certain special cases in order to sharpen this inequality. We define $c\\sb{n}$ as the smallest constant such that $c\\sb{n}\\Vert AQ - QB\\Vert \\geq \\delta\\Vert Q\\Vert$ when A, Q and B are $n \\times n$ matrices and $\\Vert \\cdot\\Vert$ is the usual norm. We then prove $c\\sb2$ = $\\sqrt{3/2}\\approx$ 1.22474, $c\\sb3$ = (8 + 5$\\sqrt{10}$)/18 $\\approx$ 1.32285, and $\\lim\\limits\\sb{n \\to \\infty}$ $c\\sb{n}$ = $\\pi$/2 $\\approx$ 1.57080. In particular, this proves $\\pi$/2 is sharp in this operator inequality.","To prove the existence of $c\\sb{sa}$ Bhatia, Davis and McIntosh first note that when A and B satisfy the given hypotheses the map ${\\cal T}:Q \\mapsto AQ - QB$ is invertible. Then they construct ${\\cal T}\\sp{-1}$ as a Fourier transform and show $\\Vert {\\cal T}\\sp{-1}\\Vert \\leq 2/\\delta$ by standard methods. Since we have $\\Vert {\\cal T}\\sp{-1}\\Vert \\leq k$ for some k if and only if $k\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert \\geq \\vert\\vert\\vert Q\\vert\\vert\\vert,$ this proves $c\\sb{sa} < 2.$ In 1987 Sz.-Nagy, referring to his earlier work, noted that if $f \\in L\\sb1$ and $\\ f(s) = s\\sp{-1}$ whenever $\\vert s\\vert$ $\\geq$ 1 then $\\Vert f\\Vert\\sb1$ $\\geq$ $\\pi$/2, and $\\pi$/2 is sharp. As a result of the original construction of ${\\cal T}\\sp{-1}$, it follows immediately that $c\\sb{sa} \\leq \\pi$/2.","For the cases when n = 2 or 3 we start with an extra assumption concerning the configuration of $\\sigma(A)$ and $\\sigma(B).$ Then we modify the Fourier transform argument to yield an upper bound on $\\Vert {\\cal T}\\sp{-1}\\Vert$. This bound is actually the correct value for $c\\sb{n}$ when n is 2 or 3, but proving it requires more work. To justify our extra assumption we formulate the problem in terms of the Schur product of two matrices, so that for each A and B there is a matrix T for which ${\\cal T}\\sp{-1}X$ = $T \\circ X$. Then we obtain estimates on $\\Vert {\\cal T}\\sp{-1}\\Vert$ from standard estimates on the Schur multiplier norm of a matrix T. From the specific information we obtain in our special case, combined with a basic inequality due to Ando, Horn and Johnson, we see that our upper bounds are correct no matter how $\\sigma(A)$ and $\\sigma(B)$ are configured. To supply lower bounds we give examples which are best possible. The ideas we've developed can then be extended to evaluate $c\\sb{sa}$. We give $n \\times n$ matrices $A\\sb{n}$, $Q\\sb{n}$ and $B\\sb{n}$ such that $\\delta\\Vert Q\\sb{n}\\Vert/\\Vert A\\sb{n}Q\\sb{n} - Q\\sb{n}B\\sb{n}\\Vert \\to \\pi/2$ as $n \\to \\infty$. This proves that $\\pi/2$ is sharp in the $AQ - QB$ inequality.","Made available in DSpace on 2011-05-07T12:27:26Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114341.pdf: 2717701 bytes, checksum: d203508a173828659a35ad1033486241 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:41:15Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:17:49-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Analysis of an inequality concerning perturbation of self-adjoint operators"]}]}],"canonical_facts":{"dc:creator":["McEachin, Raymond Vincent, Jr"],"dc:date":["2011-05-07T12:27:26Z","10000-01-01","1990"],"dc:description":["In 1983, Bhatia, Davis and McIntosh proved that if A and B are self-adjoint with dist($\\sigma(A),\\sigma(B)) \\geq \\delta$ then there is some $c\\sb{sa} < 2$ (independent of A and B) such that $c\\sb{sa}\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert \\geq \\delta\\vert\\vert\\vert Q\\vert\\vert\\vert$ for any Q and any unitary invariant norm $\\vert\\vert\\vert \\cdot\\vert\\vert\\vert$. Sz.-Nagy subsequently noted that $c\\sb{sa} \\leq \\pi$/2. It has not been known, however, if $\\pi/2$ is sharp. In this dissertation we analyze $\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert$ relative to $\\vert\\vert\\vert Q\\vert\\vert\\vert$ in certain special cases in order to sharpen this inequality. We define $c\\sb{n}$ as the smallest constant such that $c\\sb{n}\\Vert AQ - QB\\Vert \\geq \\delta\\Vert Q\\Vert$ when A, Q and B are $n \\times n$ matrices and $\\Vert \\cdot\\Vert$ is the usual norm. We then prove $c\\sb2$ = $\\sqrt{3/2}\\approx$ 1.22474, $c\\sb3$ = (8 + 5$\\sqrt{10}$)/18 $\\approx$ 1.32285, and $\\lim\\limits\\sb{n \\to \\infty}$ $c\\sb{n}$ = $\\pi$/2 $\\approx$ 1.57080. In particular, this proves $\\pi$/2 is sharp in this operator inequality.","To prove the existence of $c\\sb{sa}$ Bhatia, Davis and McIntosh first note that when A and B satisfy the given hypotheses the map ${\\cal T}:Q \\mapsto AQ - QB$ is invertible. Then they construct ${\\cal T}\\sp{-1}$ as a Fourier transform and show $\\Vert {\\cal T}\\sp{-1}\\Vert \\leq 2/\\delta$ by standard methods. Since we have $\\Vert {\\cal T}\\sp{-1}\\Vert \\leq k$ for some k if and only if $k\\vert\\vert\\vert AQ - QB\\vert\\vert\\vert \\geq \\vert\\vert\\vert Q\\vert\\vert\\vert,$ this proves $c\\sb{sa} < 2.$ In 1987 Sz.-Nagy, referring to his earlier work, noted that if $f \\in L\\sb1$ and $\\ f(s) = s\\sp{-1}$ whenever $\\vert s\\vert$ $\\geq$ 1 then $\\Vert f\\Vert\\sb1$ $\\geq$ $\\pi$/2, and $\\pi$/2 is sharp. As a result of the original construction of ${\\cal T}\\sp{-1}$, it follows immediately that $c\\sb{sa} \\leq \\pi$/2.","For the cases when n = 2 or 3 we start with an extra assumption concerning the configuration of $\\sigma(A)$ and $\\sigma(B).$ Then we modify the Fourier transform argument to yield an upper bound on $\\Vert {\\cal T}\\sp{-1}\\Vert$. This bound is actually the correct value for $c\\sb{n}$ when n is 2 or 3, but proving it requires more work. To justify our extra assumption we formulate the problem in terms of the Schur product of two matrices, so that for each A and B there is a matrix T for which ${\\cal T}\\sp{-1}X$ = $T \\circ X$. Then we obtain estimates on $\\Vert {\\cal T}\\sp{-1}\\Vert$ from standard estimates on the Schur multiplier norm of a matrix T. From the specific information we obtain in our special case, combined with a basic inequality due to Ando, Horn and Johnson, we see that our upper bounds are correct no matter how $\\sigma(A)$ and $\\sigma(B)$ are configured. To supply lower bounds we give examples which are best possible. The ideas we've developed can then be extended to evaluate $c\\sb{sa}$. We give $n \\times n$ matrices $A\\sb{n}$, $Q\\sb{n}$ and $B\\sb{n}$ such that $\\delta\\Vert Q\\sb{n}\\Vert/\\Vert A\\sb{n}Q\\sb{n} - Q\\sb{n}B\\sb{n}\\Vert \\to \\pi/2$ as $n \\to \\infty$. This proves that $\\pi/2$ is sharp in the $AQ - QB$ inequality.","Made available in DSpace on 2011-05-07T12:27:26Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114341.pdf: 2717701 bytes, checksum: d203508a173828659a35ad1033486241 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:41:15Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:17:49-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9114341","(UMI)AAI9114341","http://hdl.handle.net/2142/20056"],"dc:language":["eng"],"dc:rights":["Copyright 1990 McEachin, Raymond Vincent, Jr"],"dc:subject":["Mathematics"],"dc:title":["Analysis of an inequality concerning perturbation of self-adjoint operators"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:15Z"}