Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions

Abstract

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in C d , and L is a suitable transfer operator acting on Bergman space A 2 (Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)| ≤ A exp(−an 1/d ), where a, A > 0 are explicitly given.

A dynamical approach to accelerating numerical integration with equidistributed points

“…Even if f is rather regular, for example a polynomial, the rate at which the E M (f ) approach µ(f ) is typically no better than O(1/M ) as M → ∞. 1 The purpose of the present article is to show that this convergence can be significantly accelerated by considering linear combinations of the equidistributions E M . More precisely, for any n ∈ N we shall consider the n equidistributions…”

“…Another significant difference is that the approximation numbers a n (L t ) are O r n 1/d as n → ∞, for some 0 < r < 1. However, unlike for d = 1, we cannot use a Riemann mapping to reduce to the case where D has a simple geometry; the proof of the asymptotic a n (L t ) = O r n 1/d for general domains D is elaborated in [1]. With this stretched exponential estimate on the approximation numbers of L t , we derive the bound O θ n 1+1/d on the coefficients nc n and c n,f (0) as in Lemma 3, and hence an analogous bound on the error terms |µ(f ) − µ n (f )|.…”

Spectral structure of transfer operators for expanding circle maps

“…Thus J and hence L are trace-class. In fact, using results from [2] it is possible to show that both the singular values and the eigenvalues of L decay at an exponential rate, a property that L shares with other transfer operators arising from analytic maps (see, for example, [9,3]). …”

Resolvent Estimates for Operators Belonging to Exponential Classes

“…is compactly contained in Ω (see the previous example for the definition) then A is a compact endomorphism of L 2 Hol (Ω) and A ∈ E(a, 1/d), where a depends on the geometry of Ω and ∪ k φ k (Ω) (see [BanJ3]). …”

“…For example, if A is an integral operator with real analytic kernel given as a function on [0,1] d × [0, 1] d , then A ∈ E(a, 1/d) for some a > 0 (see [KR]). Other examples of operators in the exponential class E(a, 1/d) for some a > 0 include composition operators on Bergman spaces over domains in C d whose symbols are strict contractions, or more generally transfer operators corresponding to holomorphic map-weight systems on C d , the latter providing one of the motivations to look more closely into the properties of operators belonging to exponential classes (see Example 2.3 (iv) and [BanJ1,BanJ2,BanJ3]). …”