Abstract

Since 2007 when I was on sabbatical here at Newton Institute, I have been working on my leisure time on spectral computations and, related to that, holomorphic functional calculus which I have been calling “multicentric”. The key observation in the beginning was, informally stated: you cannot generally compute the spectrum but you can compute its complement. Making this somewhat nonsense statement exact, put me onto this path [1]. In short, multicentric calculus [2] aims to transport analysis in a complicated geometry (on the complex plain) into discs. Rather than using local variables (or conformal change of those) I introduce a new global variable which gathers information around several centers instead of just around the origin. This is a many-to-one change of variable and in this way we loose information but to compensate it we simultaneously work with several functions of the new variable. At the end of the computations the results can be transported back to the original setting. This not only opens up new computational approaches but also leads to new qualitative results, such as the extension of well known result of von Neumann (1951) on holomorphic calculus for contractions in Hilbert spaces [3]. In this talk I shall recall this extension of von Neumann’s theorem and then, if time permits, I shall discuss preliminary ideas for algebraic structures one meets in this vector valued calculus. For example, we land in a structure where vector valued functions with meromorphic components form a field. How does multiplication look like? Or derivation? How about involutions, etc.

[1] O. Nevanlinna, Computing the spectrum and representing the resolvent, Numer. Funct. Anal. Optim. 30 (9 – 10) (2009) 1025 – 1047

[2] O. Nevanlinna, Multicentric holomorphic calculus, Comput. Methods Funct. Theory 12 (1) (2012) 45 – 65.

[3] Olavi Nevanlinna: Lemniscates and K-spectral sets Journal of Functional Analysis 262 (2012) 1728 – 1741