Thomas Ransford

A primer on the Dirichlet Spaces

The Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.

Area, capacity and diameter versions of Schwarz’s Lemma

The now canonical proof of Schwarz’s Lemma appeared in a 1907 paper of Caratheodory, who attributed it to Erhard Schmidt. Since then, Schwarz’s Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a new proof of this result and extend it to include bounds on the growth of the maximum modulus. We also develop a more general approach in which the size of the image is estimated in several geometric ways via notions of radius, diameter, perimeter, area, capacity, etc.

On the norm of elementary operators

The norm problem is considered for elementary operators of the form

Computation of capacity

U_{a,b}\,{:}\,{\cal A}\,{\longrightarrow}\,{\cal A},\;x\longmapsto axb\,{+}\,bxa (a,\,b\,{\in}\,{\cal A})

Eigenvalues and power growth

in the special case when

Computation of Logarithmic Capacity

{\cal A}

Hölder Exponents of Green’s Functions of Cantor Sets

is a subalgebra of the algebra of bounded operators on a Banach space. In particular, the lower estimate

Iterated Function Systems, Capacity and Green’s Functions

\|U_{a,b}\|\geq\|a\|\|b\|

A Cartan theorem for Banach algebras

is established when the Banach space is a Hilbert space and

On mapping theorems for numerical range

{\cal A}