Generalized Halmos conjectures and constrained unitary dilations

Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hi...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Dey, Pankaj [verfasserIn] Mukherjee, Mithun

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	Higher rank numerical range Matricial range -numerical range Unitary dilation

Anmerkung:	© Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Übergeordnetes Werk:	Enthalten in: Advances in operator theory - Basel : Birkhäuser, 2016, 7(2022), 4 vom: 23. Sept.
Übergeordnetes Werk:	volume:7 ; year:2022 ; number:4 ; day:23 ; month:09

Links:	Volltext

DOI / URN:	10.1007/s43036-022-00217-x

Katalog-ID:	SPR048196525

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520			\|a Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative.
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912			\|a GBV_USEFLAG_A
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912			\|a GBV_ILN_22
912			\|a GBV_ILN_23
912			\|a GBV_ILN_24
912			\|a GBV_ILN_31
912			\|a GBV_ILN_32
912			\|a GBV_ILN_39
912			\|a GBV_ILN_40
912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_65
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
912			\|a GBV_ILN_2148
912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
912			\|a GBV_ILN_4046
912			\|a GBV_ILN_4112
912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4126
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
912			\|a GBV_ILN_4307
912			\|a GBV_ILN_4313
912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
912			\|a GBV_ILN_4325
912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4328
912			\|a GBV_ILN_4333
912			\|a GBV_ILN_4334
912			\|a GBV_ILN_4335
912			\|a GBV_ILN_4336
912			\|a GBV_ILN_4338
912			\|a GBV_ILN_4393
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allfields	10.1007/s43036-022-00217-x doi (DE-627)SPR048196525 (SPR)s43036-022-00217-x-e DE-627 ger DE-627 rakwb eng Dey, Pankaj verfasserin aut Generalized Halmos conjectures and constrained unitary dilations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. Higher rank numerical range (dpeaa)DE-He213 Matricial range (dpeaa)DE-He213 -numerical range (dpeaa)DE-He213 Unitary dilation (dpeaa)DE-He213 Mukherjee, Mithun aut Enthalten in Advances in operator theory Basel : Birkhäuser, 2016 7(2022), 4 vom: 23. Sept. (DE-627)1012767310 (DE-600)2919232-8 2538-225X nnns volume:7 year:2022 number:4 day:23 month:09 https://dx.doi.org/10.1007/s43036-022-00217-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 7 2022 4 23 09
spelling	10.1007/s43036-022-00217-x doi (DE-627)SPR048196525 (SPR)s43036-022-00217-x-e DE-627 ger DE-627 rakwb eng Dey, Pankaj verfasserin aut Generalized Halmos conjectures and constrained unitary dilations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. Higher rank numerical range (dpeaa)DE-He213 Matricial range (dpeaa)DE-He213 -numerical range (dpeaa)DE-He213 Unitary dilation (dpeaa)DE-He213 Mukherjee, Mithun aut Enthalten in Advances in operator theory Basel : Birkhäuser, 2016 7(2022), 4 vom: 23. Sept. (DE-627)1012767310 (DE-600)2919232-8 2538-225X nnns volume:7 year:2022 number:4 day:23 month:09 https://dx.doi.org/10.1007/s43036-022-00217-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 7 2022 4 23 09
allfields_unstemmed	10.1007/s43036-022-00217-x doi (DE-627)SPR048196525 (SPR)s43036-022-00217-x-e DE-627 ger DE-627 rakwb eng Dey, Pankaj verfasserin aut Generalized Halmos conjectures and constrained unitary dilations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. Higher rank numerical range (dpeaa)DE-He213 Matricial range (dpeaa)DE-He213 -numerical range (dpeaa)DE-He213 Unitary dilation (dpeaa)DE-He213 Mukherjee, Mithun aut Enthalten in Advances in operator theory Basel : Birkhäuser, 2016 7(2022), 4 vom: 23. Sept. (DE-627)1012767310 (DE-600)2919232-8 2538-225X nnns volume:7 year:2022 number:4 day:23 month:09 https://dx.doi.org/10.1007/s43036-022-00217-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 7 2022 4 23 09
allfieldsGer	10.1007/s43036-022-00217-x doi (DE-627)SPR048196525 (SPR)s43036-022-00217-x-e DE-627 ger DE-627 rakwb eng Dey, Pankaj verfasserin aut Generalized Halmos conjectures and constrained unitary dilations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. Higher rank numerical range (dpeaa)DE-He213 Matricial range (dpeaa)DE-He213 -numerical range (dpeaa)DE-He213 Unitary dilation (dpeaa)DE-He213 Mukherjee, Mithun aut Enthalten in Advances in operator theory Basel : Birkhäuser, 2016 7(2022), 4 vom: 23. Sept. (DE-627)1012767310 (DE-600)2919232-8 2538-225X nnns volume:7 year:2022 number:4 day:23 month:09 https://dx.doi.org/10.1007/s43036-022-00217-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 7 2022 4 23 09
allfieldsSound	10.1007/s43036-022-00217-x doi (DE-627)SPR048196525 (SPR)s43036-022-00217-x-e DE-627 ger DE-627 rakwb eng Dey, Pankaj verfasserin aut Generalized Halmos conjectures and constrained unitary dilations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. Higher rank numerical range (dpeaa)DE-He213 Matricial range (dpeaa)DE-He213 -numerical range (dpeaa)DE-He213 Unitary dilation (dpeaa)DE-He213 Mukherjee, Mithun aut Enthalten in Advances in operator theory Basel : Birkhäuser, 2016 7(2022), 4 vom: 23. Sept. (DE-627)1012767310 (DE-600)2919232-8 2538-225X nnns volume:7 year:2022 number:4 day:23 month:09 https://dx.doi.org/10.1007/s43036-022-00217-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 7 2022 4 23 09
language	English
source	Enthalten in Advances in operator theory 7(2022), 4 vom: 23. Sept. volume:7 year:2022 number:4 day:23 month:09
sourceStr	Enthalten in Advances in operator theory 7(2022), 4 vom: 23. Sept. volume:7 year:2022 number:4 day:23 month:09
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Higher rank numerical range Matricial range -numerical range Unitary dilation
isfreeaccess_bool	false
container_title	Advances in operator theory
authorswithroles_txt_mv	Dey, Pankaj @@aut@@ Mukherjee, Mithun @@aut@@
publishDateDaySort_date	2022-09-23T00:00:00Z
hierarchy_top_id	1012767310
id	SPR048196525
language_de	englisch
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author	Dey, Pankaj
spellingShingle	Dey, Pankaj misc Higher rank numerical range misc Matricial range misc -numerical range misc Unitary dilation Generalized Halmos conjectures and constrained unitary dilations
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topic_title	Generalized Halmos conjectures and constrained unitary dilations Higher rank numerical range (dpeaa)DE-He213 Matricial range (dpeaa)DE-He213 -numerical range (dpeaa)DE-He213 Unitary dilation (dpeaa)DE-He213
topic	misc Higher rank numerical range misc Matricial range misc -numerical range misc Unitary dilation
topic_unstemmed	misc Higher rank numerical range misc Matricial range misc -numerical range misc Unitary dilation
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title	Generalized Halmos conjectures and constrained unitary dilations
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title_full	Generalized Halmos conjectures and constrained unitary dilations
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author_browse	Dey, Pankaj Mukherjee, Mithun
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title_sort	generalized halmos conjectures and constrained unitary dilations
title_auth	Generalized Halmos conjectures and constrained unitary dilations
abstract	Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. © Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
abstractGer	Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. © Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
abstract_unstemmed	Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. © Tusi Mathematical Research Group (TMRG) 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
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title_short	Generalized Halmos conjectures and constrained unitary dilations
url	https://dx.doi.org/10.1007/s43036-022-00217-x
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author2	Mukherjee, Mithun
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doi_str	10.1007/s43036-022-00217-x
up_date	2024-07-03T17:38:41.110Z
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Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for %$k\in {\mathbb {N}},%$ the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space %${\mathcal{H}}%$ is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to %${\mathcal{H}}\oplus {\mathcal{H}}.%$ The same is true for %$k=\infty%$ provided the rank-%$\infty%$ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite (%$=N%$), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. 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