The classical interpolation theorem for the open complex unit disk, due to Nevanlinna and Pick in the early 20th century, gives an elegant criterion for the solvability of the problem as an eigenvalue problem. In the 1960s, Sarason reformulated problems of this type firmly in the language of operator theoretic function theory. This thesis will explore connections between interpolation problems on various domains (both single and several complex variables) with the viewpoint that Sarason’s work suggests. In Chapter 1, some essential preliminaries on bounded operators on Hilbert space and the functionals that act on them will be presented, with an eye on the various ways distances can be computed between operators and a certain type of ideal. The various topologies one may define on B(H) will play a prominent role in this development. Chapter 2 will introduce the concept of a reproducing kernel Hilbert space, and a distinguished operator algebra that we associate to such spaces know as the multiplier algebra. The various operator theoretic properties that multiplier algebras enjoy will be presented, with a particular emphasis on their invariant subspace lattices and the connection to distance formulae. In Chapter 3, the Nevanlinna-Pick problem will be invested in general for any repro- ducing kernel Hilbert space, with the basic heuristic for distance formulae being presented. Chapter 4 will treat a large class of reproducing kernel Hilbert spaces associated to measure spaces, where a Pick-like theorem will be established for many members of this class. This approach will closely follow similar results in the literature, including recent treatments by McCullough and Cole-Lewis-Wermer. Reproducing kernel Hilbert spaces where the analogue of the Nevanlinna-Pick theorem holds are particularly nice. In Chapter 5, the operator theory of these so-called complete Pick spaces will be developed, and used to tackle certain interpolation problems where additional constraints are imposed on the solution. A non-commutative view of interpola- tion will be presented, with the non-commutative analytic Toeplitz algebra of Popescu and Davidson-Pitts playing a prominent role. It is often useful to consider reproducing kernel Hilbert spaces which arise as natural products of other spaces. The Hardy space of the polydisk is the prime example of this. A general commutative and non-commutative view of such spaces will be presented in Chapter 6, using the left regular representation of higher-rank graphs, first introduced by Kribs-Power. A recent factorization theorem of Bercovici will be applied to these algebras, from which a Pick-type theorem may be deduced. The operator-valued Pick problem for these spaces will also be discussed. In Chapter 7, the various tools developed in this thesis will be applied to two related problems, known as the Douglas problem and the Toeplitz corona problem.

No description available.

We establish necessary conditions, in the form of the positivity of Pick-matrices, for the existence of a solution to the spectral Nevanlinna-Pick problem. We approach this problem from an operator theoretic perspective. We restate the problem as an interpolation problem on the symmetrized polydisc Γ(κ). We establish necessary conditions for a κ-tuple of commuting operators to have Γ(κ) as a complete spectral set. We then derive necessary conditions for the existence of a solution of the spectral Nevanlinna- Pick problem. The final chapter of this thesis gives an application of our results to complex geometry. We establish an upper bound for the Caratheodory distance on int Γ(κ).

The classical interpolation theorem for the open complex unit disk, due to Nevanlinna and Pick in the early 20th century, gives an elegant criterion for the solvability of the problem as an eigenvalue problem. In the 1960s, Sarason reformulated problems of this type firmly in the language of operator theoretic function theory. This thesis will explore connections between interpolation problems on various domains (both single and several complex variables) with the viewpoint that Sarason’s work suggests. In Chapter 1, some essential preliminaries on bounded operators on Hilbert space and the functionals that act on them will be presented, with an eye on the various ways distances can be computed between operators and a certain type of ideal. The various topologies one may define on B(H) will play a prominent role in this development. Chapter 2 will introduce the concept of a reproducing kernel Hilbert space, and a distinguished operator algebra that we associate to such spaces know as the multiplier algebra. The various operator theoretic properties that multiplier algebras enjoy will be presented, with a particular emphasis on their invariant subspace lattices and the connection to distance formulae. In Chapter 3, the Nevanlinna-Pick problem will be invested in general for any repro- ducing kernel Hilbert space, with the basic heuristic for distance formulae being presented. Chapter 4 will treat a large class of reproducing kernel Hilbert spaces associated to measure spaces, where a Pick-like theorem will be established for many members of this class. This approach will closely follow similar results in the literature, including recent treatments by McCullough and Cole-Lewis-Wermer. Reproducing kernel Hilbert spaces where the analogue of the Nevanlinna-Pick theorem holds are particularly nice. In Chapter 5, the operator theory of these so-called complete Pick spaces will be developed, and used to tackle certain interpolation problems where additional constraints are imposed on the solution. A non-commutative view of interpola- tion will be presented, with the non-commutative analytic Toeplitz algebra of Popescu and Davidson-Pitts playing a prominent role. It is often useful to consider reproducing kernel Hilbert spaces which arise as natural products of other spaces. The Hardy space of the polydisk is the prime example of this. A general commutative and non-commutative view of such spaces will be presented in Chapter 6, using the left regular representation of higher-rank graphs, first introduced by Kribs-Power. A recent factorization theorem of Bercovici will be applied to these algebras, from which a Pick-type theorem may be deduced. The operator-valued Pick problem for these spaces will also be discussed. In Chapter 7, the various tools developed in this thesis will be applied to two related problems, known as the Douglas problem and the Toeplitz corona problem.

Thesis (Ph. D.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves [293]-298).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2001. / Includes bibliographical references. Also available on the Internet.

This paper is a study of the properties of commutators, and deals exclusively with finite groups.

published_or_final_version / abstract / Physics / Doctoral / Doctor of Philosophy

Since Mclntosh's introduction of the H^∞-calculus for sectorial operators, the topic has been studied by many authors. Haase has constructed a similar functional calculus for strip-type operators, and has also developed an abstract framework which unifies both of these examples and more. In this thesis we use this abstract functional calculus setting to study two particular problems in operator theory. The first of these is concerned with operator sums. We ask the question of when the sum log A+log B is closed, where A and B are a pair of injective sectorial operators whose resolvents commute. We show that the sum is always closable and, when A and B are invertible, we determine sufficient conditions for the sum to be closed. These conditions are of Kalton-Weis type, and in fact ensure that AB is sectorial and that the identity log A + log B = log(AB) holds. We then identify an interpolation space on which these conditions are automatically satisfied. Our second problem is connected to the exponential of a strip-type operator B</e>, specifically the question of whether e^B is sectorial. When -1 ∈ p(e^B), the spectrum of e^B lies in a sector, and we obtain an estimate on the resolvent outside this sector. This estimate becomes closer to sectoriality as more restrictions are placed on the resolvents of B itself. This leads us to introduce the ideas of F-sectorial and F-strong strip-type operators, whose spectra are contained in a sector or strip, but which satisfy a different resolvent estimate from that of a sectorial or strong striptype operator. In some cases it is possible to define the logarithm of an F-sectorial operator or the exponential of an F-strong strip-type operator. We prove resolvent estimates for the resulting logarithms and exponentials, and explore the relationships between the various classes of operators considered.

A research report submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Science. / This research report serves as an introduction to the concept of Operator Spaces which has gained considerable momentum in its acknowledgement and research interest in the last few decades. It will highlight a very important breakthrough on the characterization of Operator spaces which occurred in the !ast few years brought about by Z.J. Ruan. It investigates the relationship of this space in relation to Banach space theory by looking at an extension theorem for linear functionals, / Andrew Chakane 2018