Draft:S-numbers

The concept of $$s$$-numbers is used in functional analysis, to describe geometric properties of a bounded linear operator $$T: X \rightarrow Y$$ acting between Banach spaces $$X$$ and $$Y$$. The concept was systematically investigated by A. Pietsch:

Definition
Let $$L(X,Y)$$ be the set of all bounded, linear operators between the Banach spaces $$X$$ and $$Y$$, and write $$\mathcal{L}:=\bigcup_{X,Y}L(X,Y)$$.

For $$T\in L(X,Y)$$ we define
 * $$\|T\|:=\|T:X\to Y\|:=\sup_{x\in X}\frac{\|Tx\|_Y}{\|x\|_X}$$,

and we often suppress the dependence on $$X$$ and $$Y$$ as they are (implicitly) specified as domain and codomain of $$T$$.

A sequence of mappings $$(s_n)_{n\in\mathbb{N}}$$ with $$s_n\colon \mathcal{L}\to\R$$ is called an $$s$$-number sequence if the following conditions are satisfied:
 * 1) $$\|T\|=s_1(T)\ge s_2(T)\ge\dots\ge 0$$,
 * 2) $$s_n(S+T)\,\le\, s_n(S)+\|T\|$$ for $$S,T\in L(X,Y)$$ and $$n\in\N$$,
 * 3) $$s_n(BTA) \,\le\, \|B\|\, s_n(T)\, \|A\|$$ for $$A\in L(X_0,X)$$, $$T\in L(X,Y)$$, $$B\in L(Y,Y_0)$$ and $$n\in\N$$,
 * 4) $$s_n(id: \ell_2^n\to\ell_2^n) =1$$ for $$n\in\N$$, where $$id(x)=x$$ is the identity, and
 * 5) $$s_n(T)=0$$ whenever $$\operatorname{rank}(T)<n$$.

We call $$s_n(T)$$ or $$s_n(T:X\to Y)$$ the $$n$$-th $$s$$-number of the operator $$T$$.

Most important s-numbers

 * The approximation numbers $$a_n(T)$$ of $$T: X\to Y$$ are defined as
 * $$a_n(T) = \inf\left\{\, \|T-L\|\colon\; L\text{ is an operator of finite rank }<n \,\right\}.$$


 * The Bernstein numbers $$b_n(T)$$ of $$T: X\to Y$$ are defined as
 * $$b_n(T) = \sup\left\{\,\inf\left\{\, \frac{\|Tx\|_Y}{\|x\|_X}\colon\, x \in V\,\right\}\colon\, \operatorname{dim}(V)=n \right\}.$$


 * The Gelfand numbers $$c_n(T)$$ of $$T: X\to Y$$ are defined as
 * $$c_n(T) = \inf\left\{\, \|T J_V^X\|\colon \,\operatorname{codim}(V)<n \,\right\},$$
 * where $$J_V^X$$ is the natural injection from $$V\subset X$$ to $$X$$.


 * The Kolmogorov numbers $$d_n(T)$$ of $$T\colon X\to Y$$ are defined as

\begin{align} d_n(T) \,&:=\, \inf\left\{\, \|Q_V^Y T\|_Y\colon\, V\subset Y,\, \operatorname{dim}(V)=n \right\} \\ &\,=\, \inf\left\{\, \sup_{x\in X}\, \inf_{y\in V}\, \|Tx-y\|_Y\colon\, V\subset Y,\, \operatorname{dim}(V)=n \right\}, \end{align}$$
 * where $$Q_V^Y$$ is the natural surjection from $$Y$$ to $$Y/V$$.

Note that all the above concepts coincide if $$X$$ and $$Y$$ are both Hilbert spaces.

Some important properties
Let us list some important property of (specific) $$s$$-numbers:


 * We have
 * $$T$$ is compact $$\iff c_n(T)\to0$$ or $$d_n(T)\to0$$.
 * The speed of convergence can therefore be used as a measure of compactness.


 * The approximation numbers $$a_n$$ are the largest $$s$$-numbers.
 * Combining properties 3.-5., we see that in fact $$s_n(T)=0 \iff {\operatorname{rank}(T)}<n$$.
 * From property 2. we see that $$|s_n(S)-s_n(T)|\le\|S-T\|$$. Hence, $$s$$-numbers are continuous in the norm topology of $$L(X,Y)$$.
 * We have $$a_n(T)=c_n(T)$$ if $$T\in L(H,Y)$$ for a Hilbert space $$H$$.
 * We have $$a_n(T)=d_n(T)$$ if $$T\in L(X,H)$$ for a Hilbert space $$H$$.
 * We have $$a_n(T)=a_n(T^*)$$ if $$T$$ is compact, and $$a_n(T^*)\le a_n(T)\le 5 a_n(T^*)$$ for all $$T\in {\mathcal{L}}$$. (Equality may fail for non-compact $$T$$. )
 * We have $$c_n(T)=d_n(T^*)$$ for all $$T\in L(X,Y)$$, and $$c_n(T^*)=d_n(T)$$ for all compact $$T\in L(X,Y)$$. (The second equality may fail for non-compact $$T$$.)