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Coloquio UAM-ICMAT

Nevanlinna-Pick Interpolation from a functional analytic viewpoint

Tirthankar Bhattacharyya (Indian Institute of Science)

Viernes, 15 de junio, Aula Naranja, ICMAT, 12:00h.

[yjsgacs id="Abstract"]
[yjsgacgroup title="Abstract" active="0"]

\[\]
About a hundred years ago, an interpolation theorem was studied by Nevanlinna (in 1919) and Pick (in 1918). It asks, given
\[small n\]
points
\[small z_1, z_2,ldots ,z_n\]
in the open unit disc
\[smallmathbb D\]
and
\[small n\]
points
\[small w_1, w_2,ldots ,w_n\]
in
\[small overline{mathbb D}\]
for some
\[small ninmathbb N\]
, when is there a holomorphic function
\[small f\]
of sup norm no more than
\[small 1\]
on the unit disc, mapping the points
\[small z_i\]
to
\[small w_i\]
? It was completely solved at that time by them, independently, using complex analysis. A landmark paper by Sarason in 1966, related this interpolation problem with functional analysis. Thus was born the celebrated Commutant Lifting Theorem. Since then, Hilbert space operator theorists have been greatly intrigued. The Nevanlinna-Pick interpolation problem has been discussed in relation to reproducing kernel Hilbert spaces in different domains. One of the ways in which it is now proved is via the so-called Realization Formula for a function in
\[small H^infty(mathbb D)\]
with sup norm no more than
\[small 1\]
. We shall see how this was ingeniously generalized to the unit bidisc by Agler and then to a very general setting by Dritschel and McCullough. The interpolation problem then takes an altogether new shape. This talk will outline this journey from the time of Navanlinna and Pick to present day state of research.[/yjsgacgroup]
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[yjsgacs id="Autor"]
[yjsgacgroup title="Tirthankar Bhattacharyya" active="0"]El profesor Tirthankar Bhattacharyya es un catedrático en el Instituto Indio de Ciencias (IISc), que es una de instituciones científicas de India de primer nivel. Ha hecho su tesis en Análisis Funcional en Instituto Estadístico de India y después de hacer postdoc en Canadá, en 2000 ingresó en IISc. Trabaja en problemas contiguos de la Teoría de Operadores y Análisis Complejo. Tiene colaboradores en muchas partes del mundo. Muchos de sus alumnos de doctorado se han convertido en investigadores de primer nivel. Tiene premios por su labor docente. Como reconocimiento de su trayectoria investigadora, fue elegido miembro de La Academia de Ciencias de India.[/yjsgacgroup]
[/yjsgacs]

Location Viernes, 15 de junio, Aula Naranja, ICMAT, 12:00h