Weighted Join Operators on Directed Trees

Abstract

Chapter 1. Background 1.1. Directed trees 1.2. Hilbert space operators 1.3. Rank one operators Prologue Chapter 2. Semigroup structures on extended directed trees 2.1. Join and meet operations on extended directed trees 2.2. A canonical decomposition of an extended directed tree Chapter 3. Weighted join operators on rooted directed trees 3.1. Closedness and boundedness 3.2. A decomposition theorem 3.3. Commutant Chapter 4. Rank one extensions of weighted join operators 4.1. Compatibility conditions and discrete Hilbert transforms 4.2. Closedness and relative boundedness 4.3. Adjoints and Gelfand-triplets 4.4. Spectral analysis Chapter 5. Special classes 5.1. Sectoriality 5.2. Normality 5.3. Symmetricity Chapter 6. Weighted Join operators on rootless directed trees 6.1. Semigroup structures on extended rootless directed trees 6.2. A decomposition theorem and spectral analysis Chapter 7. Rank one perturbations 7.1. Operator-sum 7.2. Form-sum