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We compute the rate of exponential growth of the free inverse monoid of rank r (and hence an upper bound on the corresponding rate for all r-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of $2r-1$ and $2r$, tending to $2r$ as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to $\sqrt {e(2r-1)}$ as $r \to \infty $.
We begin this chapter with a general principle. Originally, it was used in the framework of some specific spaces of analytic functions, often Hilbertian, mainly the Hardy space H2 or the Bergman space B2, but we shall apply it in many spaces, so it is worth stating as a general criterion.
In this chapter, we extend to the Orlicz spaces framework the results presented in the preceding chapters about composition operators on Hardy spaces Hp (where p ∈ [1, + ∞)) and on Bergman spaces Bp for p ≥ 1.
In this chapter, we are interested in specific examples of symbols. We already met some. For each of them, we shall sum up here their properties, even if the proofs or results appear later in this book.
In this article we introduce a gluing operation on dimer models. This allows us to construct dimer quivers on arbitrary surfaces. We study how the associated dimer and boundary algebras behave under the gluing and how to determine them from the gluing components. We also use this operation to construct homogeneous dimer quivers on annuli.
Whereas it is natural, when one studies a class of operators on a Banach or a Fréchet space, to try and describe the spectrum of elements of this class, we will devote little space to that question in this book (in the case of composition operators), essentially for three reasons:
Given a Banach space X of analytic functions on D and a symbol ϕ, and, say, w ∈ H∞, w ≠ 0, we denote by Cϕ and MwCϕ the composition and weighted composition operators
The aim of this chapter is twofold: we first give a full description of those weights β = (βn)n≥0 such that the composition operators Cϕ are bounded on the weighted Hardy space H2(β) for all symbols ϕ (analytic self-maps of D).