Rational and H^\infty dilation

Michael Dritschel (Newcastle University)

Tuesday 26th January, 2016 16:00-17:00 Maths 522


Inspired by the Sz-Nagy dilation theorem, which says that any 
contraction operator on a Hilbert space dilates to a unitary operator, 
and the two variable analogue due to Ando, Halmos asked, for what 
compact subsets of C^d is it the case that a commuting tuple of 
operators having the set as a spectral set has a dilation to a commuting 
tuple of normal operators with spectrum supported on the (distinguished) 
boundary.  Being a spectral set for a tuple means that the joint 
spectrum is contained in the set and for rational functions with poles 
off of the set, a version of the von Neumann inequality holds.

We begin by surveying known results, and speak about recent progress on 
planar domains and distinguished varieties of the bidisk.  The problem 
is related to determining whether a contractive representation of an 
analog of the disk algebra is necessarily completely contractive.  A 
similar question can be asked for H^\infty.  Even for the unit disk in 
C, this is unsettled.  We close by discussing recent progress in this case.

Some of the work discussed is joint with Daniel Estevez, Michael Jury, 
Scott McCullough, James Pickering, Batzorig Undrakh, and Dmitry Yakubovich.

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