|dc.description.abstract||The majority of this thesis is devoted to the study of Nevanlinna-Pick spaces and their
multiplier algebras. These spaces are Hilbert function spaces in which a version of the
Nevanlinna-Pick interpolation theorem from complex analysis holds. Their multiplier algebras
occupy an important place at the interface between operator algebras, operator
theory and complex analysis.
Over the last few years, the classification problem for these algebras has attracted considerable attention.
These investigations were pioneered by Davidson, Ramsey and Shalit,
who used a theorem of Agler and McCarthy to identify a given multiplier algebra with
the restriction of the multiplier algebra of the universal Nevanlinna-Pick space, namely the
Drury-Arveson space, to an analytic variety in a complex ball.
In this thesis, the classification problem is studied from three different angles. In Chapter
3, we investigate multiplier algebras associated to embedded discs in a complex ball. In
particular, we exhibit uncountably many embedded discs which are biholomorphic in a
strong sense, but whose multiplier algebras are not isomorphic. Motivated by these issues,
we use in Chapter 4 a different approach to the classification problem. Thus, we study
the spaces and their multiplier algebras directly without making use of the existence of
a universal Nevanlinna-Pick space. This allows us to completely classify the multiplier
algebras of a special class of spaces on homogeneous varieties. In Chapter 5, we investigate
the complexity of this classification problem from the point of view of Borel complexity
In Chapter 6, we show that the Hardy space on the unit disc is essentially the only
Nevanlinna-Pick space whose multiplication operators are all hyponormal.
The last part of this thesis is concerned with dilations and von Neumann’s inequality. It
has been known since the seventies that there are three commuting contractions which do
not satisfy von Neumann’s inequality. In Chapter 7, we show that every tuple of commuting
contractions which forms a multivariable weighted shift dilates to a tuple of commuting
unitaries and hence satisfies von Neumann’s inequality, thereby providing a positive answer
to a question of Shields and Lubin from 1974.||en