{"id":"https://openalex.org/W7220728692","doi":"https://doi.org/10.48550/arxiv.2610.04583","title":"Simultaneous Williamson's normal form for positive operators","display_name":"Simultaneous Williamson's normal form for positive operators","publication_year":2026,"publication_date":"2026-10-03","ids":{"openalex":"https://openalex.org/W7220728692","doi":"https://doi.org/10.48550/arxiv.2610.04583"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.04583","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04583","pdf_url":"https://arxiv.org/pdf/2610.04583","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.04583","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5092838197","display_name":"Rudra R. Kamat","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kamat, Rudra","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5025897566","display_name":"Anmary Tonny","orcid":"https://orcid.org/0009-0006-8342-1979","observed_orcids":["https://orcid.org/0009-0006-8342-1979"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tonny, Anmary","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.6528000235557556,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.6528000235557556,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10884","display_name":"Holomorphic and Operator Theory","score":0.147599995136261,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.06759999692440033,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/positive-operators","display_name":"positive operators","score":0.9259999990463257},{"id":"https://openalex.org/keywords/simultaneous-diagonalization","display_name":"simultaneous diagonalization","score":0.9010000228881836},{"id":"https://openalex.org/keywords/spectral-invariants","display_name":"spectral invariants","score":0.906000018119812},{"id":"https://openalex.org/keywords/unbounded-operators","display_name":"unbounded operators","score":0.4740000069141388}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7886999845504761},{"id":"https://openalex.org/C168619227","wikidata":"https://www.wikidata.org/wiki/Q2190991","display_name":"Symplectic geometry","level":2,"score":0.7505000233650208},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.6470000147819519},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5856999754905701},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.536899983882904},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5213000178337097},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.47589999437332153},{"id":"https://openalex.org/C43929395","wikidata":"https://www.wikidata.org/wiki/Q1198874","display_name":"Operator theory","level":2,"score":0.3849000036716461},{"id":"https://openalex.org/C60064506","wikidata":"https://www.wikidata.org/wiki/Q1262627","display_name":"Rigged Hilbert space","level":4,"score":0.37720000743865967},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33180001378059387},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.31619998812675476},{"id":"https://openalex.org/C130190758","wikidata":"https://www.wikidata.org/wiki/Q1064500","display_name":"Symplectic manifold","level":3,"score":0.29840001463890076},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C174058634","wikidata":"https://www.wikidata.org/wiki/Q5155314","display_name":"Compact operator on Hilbert space","level":4,"score":0.28049999475479126},{"id":"https://openalex.org/C35958075","wikidata":"https://www.wikidata.org/wiki/Q1425077","display_name":"Spectral theorem","level":3,"score":0.27790001034736633},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.2721000015735626},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2718000113964081},{"id":"https://openalex.org/C2778975207","wikidata":"https://www.wikidata.org/wiki/Q7078836","display_name":"Of the form","level":2,"score":0.25690001249313354},{"id":"https://openalex.org/C2985906921","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Spectral properties","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.04583","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04583","pdf_url":"https://arxiv.org/pdf/2610.04583","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.04583","doi":"https://doi.org/10.48550/arxiv.2610.04583","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.04583","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.04583","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04583","pdf_url":"https://arxiv.org/pdf/2610.04583","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322211","display_name":"Indian Institute of Technology Madras","ror":"https://ror.org/03v0r5n49"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220728692.pdf","grobid_xml":"https://content.openalex.org/works/W7220728692.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W6891920951"],"related_works":["https://openalex.org/W2797622153","https://openalex.org/W2516301525","https://openalex.org/W4293230775","https://openalex.org/W4386008173","https://openalex.org/W2027030816","https://openalex.org/W4385966254","https://openalex.org/W4405035030","https://openalex.org/W2951781473","https://openalex.org/W2761533816","https://openalex.org/W4249283750"],"abstract_inverted_index":{"We":[0],"prove":[1],"a":[2,16,24,30,60,75],"simultaneous":[3],"Williamson's":[4,67],"normal":[5,68],"form":[6,69],"for":[7,79],"two":[8],"symplectically":[9],"commuting":[10],"positive":[11],"operators":[12],"$A,":[13],"B$":[14],"on":[15],"real":[17],"symplectic":[18,43,62],"Hilbert":[19],"space":[20],"$(\\mathbf{H},":[21],"J)$.":[22],"Under":[23],"natural":[25],"commutativity":[26],"condition,":[27],"there":[28],"exists":[29],"(generally":[31],"unbounded)":[32],"operator":[33],"$\u03a6$":[34],"that":[35],"simultaneously":[36],"diagonalizes":[37],"$A$":[38],"and":[39,56,73],"$B$":[40],"in":[41],"the":[42,80],"setting":[44],"as":[45],"$A":[46],"=":[47,52],"\u03a6^t":[48,53],"M_A":[49],"\u03a6$,":[50,55],"$B":[51],"M_B":[54],"maps":[57],"$J$":[58],"to":[59,70],"canonical":[61],"operator.":[63],"The":[64],"result":[65],"generalizes":[66],"infinite":[71],"dimensions":[72],"yields":[74],"complete":[76],"spectral":[77],"invariant":[78],"pair":[81],"$(A,B)$.":[82]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}