Entropy number

In functional analysis, entropy numbers are indicators of continuous linear operators . The concept is based on the term epsilon entropy .

definition

External entropy numbers

Let be and Banach spaces and a linear continuous operator , that's what one calls ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T}$ ${\displaystyle T\in L(X,Y)}$

${\displaystyle \varepsilon _{n}(T):=\inf \left\{\varepsilon >0|\exists x_{1},\dots ,x_{n}\in Y{\text{mit }}T(B_{X})\subseteq \bigcup _{i=1}^{n}{{x_{i}}+\varepsilon B_{Y}}\right\}}$

nth entropy number of T, where and are the closed unit spheres in X and Y , respectively . We call ${\displaystyle B_{X}}$ ${\displaystyle B_{Y}}$

${\displaystyle e_{n}:=\varepsilon _{2^{n}-1}(T)}$

the nth dyadic Entropiezahl of T .

In the transition from the “normal” entropy numbers to the dyadic ones, no essential information is lost in the asymptotic view. That is why the dyadic entropy numbers are often just called entropy numbers.

Inner entropy numbers

Let be and Banach spaces and a linear continuous operator , that's what one calls ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T}$ ${\displaystyle T\in L(X,Y)}$

${\displaystyle \varphi _{n}(T):=\sup \left\{\rho >0:\exists \,p>n{\text{ mit }}y_{1},\dots ,y_{p}\in T(B_{X}),\,\left\|y_{i}-y_{j}\right\|\geq 2\rho \,\forall i\neq j\right\}}$

inner Entropiezahl of T .

${\displaystyle f_{n}(T):=\varphi _{2^{n}-1}(T)}$

is dyadic inner Entropiezahl of T mentioned.

Relationship between inner and outer entropy numbers

As Carl and Stephani have shown in their book Entropy, compactness and the approximation of operators , the relationship exists

${\displaystyle \varphi _{n}(T)\leq \varepsilon _{n}(T)\leq 2\varphi _{n}(T)}$

which is why one usually only looks at it. ${\displaystyle e_{n}(T)}$

comment

If you look at the definition of the entropy numbers, you can see the following elementary relationship:

${\displaystyle T}$ is compact ${\displaystyle \Leftrightarrow e_{n}(T)\rightarrow 0.}$

On the basis of this fact, the entropy numbers can be used to assign a "degree of compactness" to the operator, i. H. the faster the entropy numbers fall towards 0, the more compact the operator.

literature

• Hermann König : Eigenvalue Distribution of Compact Operators , Birkhäuser, 1985 (contains a good introduction to the theory of s-numbers)
• David Eric Edmunds , Hans Triebel : Function Spaces, Entropy Numbers, Differential Operators , Cambridge University Press, 1994
• Bernd Carl, Irmtraud Stephani: Entropy, compactness and the approximation of operators , Cambridge University Press, 1990