The failure of rational dilation on a triply connected domain
By Michael A. Dritschel, Scott McCullough. 2003
73 citations recorded by the source corpus.
Let R denote a domain in C with boundary B. Let X denote the closure of R. An operator T on a complex Hilbert space H has X as a spectral set if cr(T) C X and ||/(T)||<||/||n = sup{|/(*)|:*eJR} for every rational function / with poles off X. The expression f(T) may be inter preted either in terms of the Riesz functional calculus, or simply by writing the ra tional function / as pq~~l for polynomials p and q, in which case f(T) = p(T)q(T)~l. The operator T has a normal i?-dilation if there exists a Hilbert space /C con taining H and a normal operator TV on K so that
Subfields: Applied Mathematics, Mathematical Physics
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Intersections of several disks of the Riemann sphere as K-spectral sets (2008, background): “However, the analogue of Agler’s result is not true for triply connected domains (s ee [2, 10 , 18]).”
- Operator Theory on Symmetrized Bidisc (2012, background): “In particular, it is known that the normal ∂K-dilation holds if K is the closure of an annulus [2] and fails when K is a triply connected domain in C [14].”
- The tetrablock as a spectral set (2014, background): “omain. Whether a domain in the plane with a single hole enjoyed the property P was open for a long time till Agler showed in [3] that the annulus has this property. Dritschel and McCullough proved in [15] that a multiply connected domain in general does not. They use Gel’fand-Naimark-Segal construct”