S number function

An s-number function is a mapping s commonly used in functional analysis , which assigns a sequence with the following properties to each operator for Banach spaces E and F : ${\displaystyle T\in {\mathcal {L}}(E,F)}$ ${\displaystyle (s_{n}(T))}$

1. Monotony: ${\displaystyle \|T\parallel =s_{1}(T)\geq s_{2}(T)\geq ...\geq 0,\;T\in {\mathcal {L}}(E,F)}$
2. Additivity: for ${\displaystyle s_{n}(S+T)\leq s_{n}(S)+\|T\parallel }$ ${\displaystyle S,T\in {\mathcal {L}}(E,F)}$
3. Ideal property: ${\displaystyle s_{n}(RST)\leq \left\|R\right\|s_{n}(S)\|T\parallel ,\;T\in {\mathcal {L}}(E_{1},E),S\in {\mathcal {L}}(E,F),R\in {\mathcal {L}}(F,F_{1})}$
4. Range property: for with ${\displaystyle s_{n}(T)=0}$ ${\displaystyle T\in {\mathcal {F}}(E,F)}$ ${\displaystyle \operatorname {rang} \,T<n}$
5. Normalization: ${\displaystyle s_{n}(id:{l_{2}}^{n}\rightarrow {l_{2}}^{n})=1}$

The value is called n th s number of T called. ${\displaystyle s_{n}(T)}$

The approximation numbers, the Gelfand numbers, the Kolmogorow numbers, the Weyl numbers and the Tichomirov numbers are additive s-number functions. The most prominent representatives of the pseudo-s-number functions are the entropy numbers .

literature

• Hermann König: Eigenvalue distribution of compact operators . In: Integral Equations and Operator Theory . tape 9 , no. 4 . Birkhauser Verlag, July 1986, ISSN  0378-620X , p. 610–612 , doi : 10.1007 / BF01204633 (contains an introduction to the theory of s-numbers).

Individual evidence

1. ^ Albrecht Pietsch : Operator ideals. VEB Deutscher Verlag der Wissenschaft, 1978.