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SUMMARY:MULTICENTRIC CALCULUS\, What and Why? - Olavi Nevanlinna (Aalto Un
 iversity)
DTSTART:20130110T150000Z
DTEND:20130110T160000Z
UID:TALK42519@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION:Since 2007 when I was on sabbatical here at Newton Institute\,
  I have been working\non my leisure time on spectral computations and\, re
 lated to that\, holomorphic functional\ncalculus which I have been calling
  "multicentric".\nThe key observation in the beginning was\, informally st
 ated: you cannot generally\ncompute the spectrum but you can compute its c
 omplement. Making this somewhat\nnonsense statement exact\, put me onto th
 is path [1].\nIn short\, multicentric calculus [2] aims to transport analy
 sis in a complicated geometry\n(on the complex plain) into discs. Rather t
 han using local variables (or conformal\nchange of those) I introduce a ne
 w global variable which gathers information around\nseveral centers instea
 d of just around the origin. This is a many-to-one change of variable\nand
  in this way we loose information but to compensate it we simultaneously w
 ork\nwith several functions of the new variable. At the end of the computa
 tions the results\ncan be transported back to the original setting.\nThis 
 not only opens up new computational approaches but also leads to new quali
 tative\nresults\, such as the extension of well known result of von Neuman
 n (1951) on\nholomorphic calculus for contractions in Hilbert spaces [3].\
 nIn this talk I shall recall this extension of von Neumann’s theorem and
  then\, if\ntime permits\, I shall discuss preliminary ideas for algebraic
  structures one meets in\nthis vector valued calculus. For example\, we la
 nd in a structure where vector valued\nfunctions with meromorphic componen
 ts form a field.\nHow does multiplication look like?\nOr derivation? How a
 bout involutions\, etc.\n\n\n[1] O. Nevanlinna\, Computing the spectrum an
 d representing the resolvent\, Numer. Funct. Anal. Optim. 30 (9 - 10) (200
 9) 1025 - 1047\n\n[2] O. Nevanlinna\, Multicentric holomorphic calculus\, 
 Comput. Methods Funct.\nTheory 12 (1) (2012) 45 - 65.\n\n[3] Olavi Nevanli
 nna: Lemniscates and K-spectral sets Journal of Functional Analysis 262 (2
 012) 1728 - 1741
LOCATION:MR 14\, CMS
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