\[\]
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
\[\small \mathbb D\]
and
\[\small n\]
points
\[\small w_1, w_2, \ldots ,w_n\]
in
\[\small \overline{\mathbb D}\]
for some
\[\small n \in \mathbb N\]
, 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.