The failure of rational dilation on the tetrablock
We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for whic...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Pal, Sourav [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2015transfer abstract |
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| Umfang: |
22 |
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| Übergeordnetes Werk: |
Enthalten in: Corrigendum to “Rifampicin resistance mutations in the rpoB gene of - Urusova, Darya V. ELSEVIER, 2022, Amsterdam [u.a.] |
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| Übergeordnetes Werk: |
volume:269 ; year:2015 ; number:7 ; day:1 ; month:10 ; pages:1903-1924 ; extent:22 |
| Links: |
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| DOI / URN: |
10.1016/j.jfa.2015.07.006 |
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ELV024005509 |
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| 520 | |a We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. | ||
| 520 | |a We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. | ||
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10.1016/j.jfa.2015.07.006 doi GBVA2015022000016.pica (DE-627)ELV024005509 (ELSEVIER)S0022-1236(15)00284-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Pal, Sourav verfasserin aut The failure of rational dilation on the tetrablock 2015transfer abstract 22 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. 47A15 Elsevier 47A13 Elsevier 47A25 Elsevier 47A45 Elsevier 47A20 Elsevier Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:269 year:2015 number:7 day:1 month:10 pages:1903-1924 extent:22 https://doi.org/10.1016/j.jfa.2015.07.006 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 269 2015 7 1 1001 1903-1924 22 045F 510 |
| spelling |
10.1016/j.jfa.2015.07.006 doi GBVA2015022000016.pica (DE-627)ELV024005509 (ELSEVIER)S0022-1236(15)00284-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Pal, Sourav verfasserin aut The failure of rational dilation on the tetrablock 2015transfer abstract 22 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. 47A15 Elsevier 47A13 Elsevier 47A25 Elsevier 47A45 Elsevier 47A20 Elsevier Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:269 year:2015 number:7 day:1 month:10 pages:1903-1924 extent:22 https://doi.org/10.1016/j.jfa.2015.07.006 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 269 2015 7 1 1001 1903-1924 22 045F 510 |
| allfields_unstemmed |
10.1016/j.jfa.2015.07.006 doi GBVA2015022000016.pica (DE-627)ELV024005509 (ELSEVIER)S0022-1236(15)00284-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Pal, Sourav verfasserin aut The failure of rational dilation on the tetrablock 2015transfer abstract 22 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. 47A15 Elsevier 47A13 Elsevier 47A25 Elsevier 47A45 Elsevier 47A20 Elsevier Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:269 year:2015 number:7 day:1 month:10 pages:1903-1924 extent:22 https://doi.org/10.1016/j.jfa.2015.07.006 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 269 2015 7 1 1001 1903-1924 22 045F 510 |
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10.1016/j.jfa.2015.07.006 doi GBVA2015022000016.pica (DE-627)ELV024005509 (ELSEVIER)S0022-1236(15)00284-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Pal, Sourav verfasserin aut The failure of rational dilation on the tetrablock 2015transfer abstract 22 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. 47A15 Elsevier 47A13 Elsevier 47A25 Elsevier 47A45 Elsevier 47A20 Elsevier Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:269 year:2015 number:7 day:1 month:10 pages:1903-1924 extent:22 https://doi.org/10.1016/j.jfa.2015.07.006 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 269 2015 7 1 1001 1903-1924 22 045F 510 |
| allfieldsSound |
10.1016/j.jfa.2015.07.006 doi GBVA2015022000016.pica (DE-627)ELV024005509 (ELSEVIER)S0022-1236(15)00284-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Pal, Sourav verfasserin aut The failure of rational dilation on the tetrablock 2015transfer abstract 22 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. 47A15 Elsevier 47A13 Elsevier 47A25 Elsevier 47A45 Elsevier 47A20 Elsevier Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:269 year:2015 number:7 day:1 month:10 pages:1903-1924 extent:22 https://doi.org/10.1016/j.jfa.2015.07.006 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 269 2015 7 1 1001 1903-1924 22 045F 510 |
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failure of rational dilation on the tetrablock |
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The failure of rational dilation on the tetrablock |
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We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. |
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We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. |
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We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. |
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