Page 72 - Textos de Matemática Vol. 34

62 SUSANA D. MOURA
5. Entropy numbers
The aim of this section is to present an estimation for the entropy numbers of
(s,Ψa )
some compact embeddings between Besov spaces Bp,q (Γ) on a (d, Ψ)-set Γ, which
plays a decisive role in the sequel.
First of all, we briefly recall some basic facts concerning entropy numbers. Let
A1 and A2 be complex quasi-Banach spaces and let T be a bounded linear map from A1 into A2. If T is compact then for any given ε > 0 there are finitely many balls in A2 of radius ε which cover the image T (UA1 ) of the unit ball UA1 := {a ∈ A1 : ∥a|A1∥ ≤ 1}.
Definition 5.1. For each k ∈ N, the kth entropy number ek(T) of T is the infimum of all numbers ε > 0 such that there exist 2k−1 balls in A2 of radius ε which cover T (UA1 ).
Note that (ek (T ))k∈N is a non-increasing sequence of non-negative numbers which is bounded from above by ∥T∥ and, moreover, limk→∞ ek(T) = 0 if and only if T is compact. If S denotes another bounded linear map from A2 into an- other quasi-Banach space, then it holds ek+l−1(S◦T) ≤ ek(S)el(T) for all k,l ∈ N; this is the so called multiplicative property of entropy numbers.
For details and further properties on entropy numbers we refer to [10], [14] and [25] (restricted to the case of Banach spaces), and to [15] for some extensions to quasi-Banach spaces.
An important application of entropy numbers is to spectral theory. If T is a compact operator on the quasi-Banach space A, then the spectrum of T , apart from the point 0, consists solely of an at most countable infinite number of eigenvalues of finite algebraic multiplicity. We denote by μk(T), k ∈ N, the non-zero eigenvalues of T, repeated according to algebraic multiplicity and ordered by non-increasing modulus. If T has only finitely many distinct such eigenvalues and M is the sum of their algebraic multiplicities, we set μk (T ) = 0 for all k > M . An excellent link between the eigenvalues of T and its entropy numbers is provided by Carl’s inequality
in particular,
k 1/k |μm(T)|
m=1
≤ inf 2n/2k en(T) , n∈N
√
k ∈ N,
|μk(T)| ≤
2 ek(T).
This result was originally proved by Carl in [12] and Carl and Triebel in [11] when A is a Banach space. An extension to quasi-Banach spaces can be found in [15, Theorem 1.3.4, p. 18].