TUM Library

This handbook concerns the subject of holomorphic function spaces and operators acting on them. Topics include Bergman spaces, Hardy spaces, Besov/Sobolev spaces, Fock spaces, and the space of Dirichlet series. Operators discussed in the book include Toeplitz operators, Hankel operators, composition operators, and Cowen-Douglas class operators

2.5 Existence of nonsubnormal hyponormal 2-variable weighted shifts2.6 Propagation in the 2-variable hyponormal case; 2.7 A measure-theoretic necessary (but not sufficient!) condition for the existence of a lifting; 2.8 Reconstruction of the Berger measure for 2-variable weighted shifts whose core is of tensor form; 2.9 The subnormal completion problem for 2-variable weighted shifts; 2.10 Spectral picture of hyponormal 2-variable weighted shifts; 2.11 A bridge between 2-variable weighted shifts and shifts on directed trees; 2.12 The spherical Aluthge transform; References

3: Commutants, Reducing Subspaces, and von Neumann Algebras3.1 Introduction; 3.2 Commutants and reducing subspaces for multiplication operators on the Hardy space H2(D); 3.3 The case of the Bergman space L2a(D); 3.4 The case of Bergman space over a polygon; 3.5 The case of the Bergman space over high dimensional domains; 3.6 Further questions; References; 4: Operators in the Cowen-Douglas Class and Related Topics; 4.1 Introduction; 4.2 Some future directions and further thoughts; References; 5: Toeplitz Operators and Toeplitz C*-Algebras; 5.1 Introduction

5.2 Toeplitz operators on Hilbert spaces of multi-variable holomorphic functions5.3 Strongly pseudoconvex domains; 5.4 Symmetric domains and Jordan triples; 5.5 Holomorphic function spaces on symmetric domains; 5.6 Toeplitz C*-algebras on symmetric domains; 5.7 Hilbert quotient modules and Kepler varieties; 5.8 Toeplitz operators on Reinhardt domains; References; 6: Möbius Invariant Zp and ZK Spaces; 6.1 Introduction; 6.2 Background; 6.3 Basic properties of Zp spaces; 6.4 Carleson measures; 6.5 The boundary value characterizations; 6.6 ZK spaces; 6.7 K-Carleson measures

6.8 Boundary ZK spaces6.9 Composition operators on Zp and ZK spaces; References; 7: Analytical Aspects of the Drury-Arveson Space; 7.1 Introduction; 7.2 von Neumann inequality for row contractions; 7.3 The multipliers; 7.4 A family of reproducing-kernel Hilbert spaces; 7.5 Essential normality; 7.6 Expanding on Drury's idea; 7.7 Closure of the polynomials; References; 8: A Brief Survey of Operator Theory in H2(D2); 8.1 Introduction; 8.2 Background; 8.3 Nagy-Foias theory in H2(D2); 8.4 Commutators; 8.5 Two-variable Jordan block; 8.6 Fredholmness of the pairs (R1,R2) and (S1,S2)

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