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The failure of rational dilation on the tetrablock.
- Source :
-
Journal of Functional Analysis . Oct2015, Vol. 269 Issue 7, p1903-1924. 22p. - Publication Year :
- 2015
-
Abstract
- 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00221236
- Volume :
- 269
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Journal of Functional Analysis
- Publication Type :
- Academic Journal
- Accession number :
- 108823211
- Full Text :
- https://doi.org/10.1016/j.jfa.2015.07.006