Inequalities between s-numbers

Mario Ullrich Institut für Analysis, Johannes Kepler Universität Linz, Austria [email protected] In memory of Albrecht Pietsch

(Date: September 16, 2014)

Abstract.

Singular numbers of linear operators between Hilbert spaces were generalized to Banach spaces by s-numbers (in the sense of Pietsch). This allows for different choices, including approximation, Gelfand, Kolmogorov and Bernstein numbers. Here, we present an elementary proof of a bound between the smallest and the largest s-number.

Key words and phrases:

s-numbers, Hilbert numbers, Pietsch

1991 Mathematics Subject Classification:

Primary 47B06; Secondary 46B50, 47B01

We start with introducing the terminology and a presentation of the results. In Section 2 we will discuss them and some history. Proofs are given in Section 3.

1. s-numbers

In what follows, let $X$ , $Y$ , $Z$ and $W$ be real or complex Banach spaces. The (closed) unit ball of $X$ is denoted by $B_{X}$ , the dual space of $X$ by $X^{\prime}$ , and the identity map on $X$ is denoted by $I_{X}$ . For a closed subspace $M\subset X$ , we write $J_{M}^{X}$ for the embedding $J_{M}^{X}\colon M\to X$ with $J_{M}^{X}(x)=x$ , and $Q_{M}^{X}$ for the canonical map $Q_{M}^{X}\colon X\to X/M$ with $Q_{M}^{X}(x)=x+M$ onto the quotient space $X/M:=\{x+M\colon x\in X\}$ with norm $\|x+M\|_{X/M}:=\inf_{m\in M}\|x+m\|_{X}$ . The dimension of a subspace $M\subset X$ is denoted by $\dim(M)$ , and by $\operatorname{codim}(M):=\dim(X/M)$ we denote its codimension.

The class of all bounded linear operators between Banach spaces is denoted by $\mathcal{L}$ , and by $\mathcal{L}(X,Y)$ we denote those operators from $X$ to $Y$ , equipped with the operator norm. The rank of an operator $S\in\mathcal{L}(X,Y)$ is defined by $\operatorname{rank}(S):=\dim(S(X))$ .

A map $S\to(s_{n}(S))_{n\in\mathbb{N}}$ assigning to every operator $S\in\mathcal{L}$ a non-negative scalar sequence $(s_{n}(S))_{n\in\mathbb{N}}$ is called an s-number sequence if, for all $n\in\mathbb{N}$ , the following conditions are satisﬁed

(S1)

$\|S\|=s_{1}(S)\geq s_{2}(S)\geq\dots\geq 0$ for all $S\in\mathcal{L}$ ,
(S2)

$s_{n}(S+T)\,\leq\,s_{n}(S)+\|T\|$ for all $S,T\in\mathcal{L}(X,Y)$ ,
(S3)

$s_{n}(BSA)\,\leq\,\|B\|\,s_{n}(S)\,\|A\|$ where $W\stackrel{{\scriptstyle A}}{{\longrightarrow}}X\stackrel{{\scriptstyle S}}{{\longrightarrow}}Y\stackrel{{\scriptstyle B}}{{\longrightarrow}}Z$ ,
(S4)

$s_{n}(I_{\ell_{2}^{n}})=1$ ,
(S5)

$s_{n}(S)=0$ whenever $\operatorname{rank}(S)<n$ .

We call $s_{n}(S)$ the $n^{\rm th}$ s-number of the operator $S$ . To indicate the underlying Banach spaces, we sometimes write $s_{n}(S\colon X\to Y)$ .

There are some especially important examples of s-numbers:

•

approximation numbers:

$a_{n}(S)\,:=\,\inf\Bigl{\{}\|S-L\|\colon L\in\mathcal{L}(X,Y),\;\operatorname{rank}(L)<n\Bigr{\}}$
•

Bernstein numbers:

$b_{n}(S)\,:=\,\sup\Bigl{\{}\inf_{x\in M\setminus\{0\}}\frac{\|Sx\|}{\|x\|}\colon M\subset X,\;\dim(M)=n\Bigr{\}}$
•

Gelfand numbers:

$c_{n}(S)\,:=\,\inf\Bigl{\{}\|SJ_{M}^{X}\|\colon M\subset X,\;\operatorname{codim}(M)<n\Bigr{\}}$
•

Kolmogorov numbers:

$d_{n}(S)\,:=\,\inf\Bigl{\{}\|Q_{N}^{Y}S\|\colon N\subset Y,\;\dim(N)<n\Bigr{\}}$
•

Weyl numbers:

$x_{n}(S)\,:=\,\sup\Biggl{\{}\frac{a_{n}(SA)}{\|A\|}\colon A\in\mathcal{L}(\ell_{2},X),\;A\neq 0\Biggr{\}}$

•

Hilbert numbers:

\begin{split}h_{n}(S)\,:&=\,\sup\Biggl{\{}\frac{a_{n}(BSA)}{\|B\|\|A\|}\colon A\in\mathcal{L}(\ell_{2},X),\;B\in\mathcal{L}(Y,\ell_{2}),\;A,B\neq 0\Biggr{\}}.\\ \end{split}

We refer to [13, 14] for a detailed treatment of the above, and a few other, s-numbers and their specific properties.

Remark 1.

The original definition of s-numbers in [11] used the stronger norming property $(S4^{\prime})\colon s_{n}(I_{X})=1$ for all $X$ with $\dim(X)\geq n$ . This did not allow for $x_{n}$ and $h_{n}$ , and has been weakened in [1, 12] for defining them, leading to the least restrictive axioms that still imply uniqueness for Hilbert spaces, see Proposition 2. It is sometimes assumed that the s-number sequence is additive, i.e., $(S2^{\prime})\colon s_{m+n-1}(S+T)\leq s_{m}(S)+s_{n}(T)$ , or multiplicative, i.e., $(S3^{\prime})\colon s_{m+n-1}(ST)\leq s_{m}(S)s_{n}(T)$ . Both properties hold for $a_{n},c_{n},d_{n},x_{n}$ , while $h_{n}$ is only additive, and $b_{n}$ is neither of the two, see [14, 15].

The following proposition is well-known, see [13, 2.3.4 & 2.6.3 & 2.11.9].

Proposition 2.

For every s-number sequence $(s_{n})$ , $S\in\mathcal{L}$ and $n\in\mathbb{N}$ , we have

h_{n}(S)\,\leq\,s_{n}(S)\,\leq\,a_{n}(S).

Equalities hold if $S\in\mathcal{L}(H,K)$ for Hilbert spaces $H$ and $K$ .

For convenience, we present a sketch of the proof of the inequalities in Section 3. Using only elementary arguments, we prove the following reverse inequality, which is known in a more involved form based on an intermediate comparison with $x_{n}(S)$ , see [13, 2.10.7] or [14, 6.2.3.14], or Remark 6 in [6].

Theorem 3.

For all $S\in\mathcal{L}$ and $n\in\mathbb{N}$ , we have

\max\Bigl{\{}c_{n}(S),\,d_{n}(S)\Bigr{\}}\;\leq\;n\,\left(\prod_{k=1}^{n}h_{k}(S)\right)^{1/n}.

Since $c_{n}(I\colon\ell_{1}\to\ell_{\infty})\asymp 1$ and $h_{n}(I\colon\ell_{1}\to\ell_{\infty})\asymp n^{-1}$ , see [14, 6.2.3.14] and [3], this result is best possible up to constants.

We cannot obtain bounds for individual $n$ from Theorem 3, see also Remark 5, but combining it with the inequality $n^{\alpha}\leq e^{\alpha}\,(n!)^{\alpha/n}$ for $\alpha\geq 0$ , we obtain a more handy form. Moreover, the known fact that $a_{n}(S)\leq(1+\sqrt{n})\,c_{n}(S)$ , see [13, 2.10.2], leads to a bound between $a_{n}$ and $h_{n}$ , and hence between all s-numbers.

Corollary 4.

For all $S\in\mathcal{L}$ , $\alpha>0$ and $n\in\mathbb{N}$ , we have

c_{n}(S)\,\leq\,e^{\alpha}\,n^{-\alpha+1}\cdot\sup_{k\leq n}\,k^{\alpha}\,h_{k}(S),

and

a_{n}(S)\,\leq\,2\,e^{\alpha}\,n^{-\alpha+3/2}\cdot\sup_{k\leq n}\,k^{\alpha}\,h_{k}(S).

2. A bit of history

We provide a brief description of the relevant facts from (the highly recommended) “History of Banach Spaces and Linear Operators” of Pietsch [14], and give some further references.

Singular numbers of operators on Hilbert spaces have become fundamental tools in (applied) mathematics since their introduction in 1907 by Schmidt [18]. For compact $S\in\mathcal{L}(H,K)$ between complex Hilbert spaces $H,K$ , the singular numbers are defined by $s_{k}(S):=\sqrt{\lambda_{k}(SS^{*})}$ , where the eigenvalues $\lambda_{k}(T)$ of $T\in\mathcal{L}(X,X)$ are characterized by $Te_{k}=\lambda_{k}(T)\cdot e_{k}$ for some $e_{k}\in X\setminus\{0\}$ , and ordered decreasingly. Applications range from the study of eigenvalue distributions of operators, see [5] or [14, 6.4], to the classification of operator ideals [14, 6.3], to the singular value decomposition, aka Schmidt representation, with its many applications.

s-numbers are a generalization to linear operators between Banach spaces. However, there is no unique substitute for singular numbers but, depending on the context, different s-numbers may be used to quantify compactness, while others may be easier to compute. As Pietsch wrote in [14, 6.2.2.1], “we have a large variety of s-numbers that make our life more interesting.”

Most notably, $a_{n}$ , $b_{n}$ , $c_{n}$ and $d_{n}$ were already known in the 1960s, sometimes in a related form as “width” of a set, see [4, 20] or [14, 6.2.6], and are by now part of the foundation of approximation theory [17] and information-based complexity [8, 10]. More recent treatments of the subject and extensions can be found, e.g., in [2, 7, 19]. Let us also highlight [6], where we discuss the relation of s-numbers to minimal approximation errors in detail, and use variants of Theorem 3 to bound the maximal gain of randomized/adaptive algorithms over deterministic/non-adaptive ones.

An axiomatic theory of s-numbers has been developed by Pietsch [11, 13, 14] in the 1970s. This, in particular, allowed for a characterization of the smallest/largest s-number (with certain properties), see also Remark 1.

Inequalities and several relations between s-numbers have already been collected in [11], see also [13, 2.10] and [14, 6.2.3.14]. A particularly interesting bound is $d_{n}(S)\leq n^{2}b_{n}(S)$ , which was proven by Mityagin and Henkin [9] in 1963. They also conjectured that $n^{2}$ can be replaced by $n$ , see also [17, p. 24]. This bound with $b_{n}$ replaced by $h_{n}$ , and the corresponding conjecture, have been given in [1], and apparently, the bound has not been improved since then. However, in the weaker form as in Theorem 3, this problem has been solved by Pietsch [12] in 1980, see the final remarks there. In particular, it is shown that there is some $C_{\alpha}>0$ such that $\sup_{k\geq 1}k^{\alpha}d_{k}(S)\leq C_{\alpha}\,\sup_{k\geq 1}k^{\alpha+1}h_{k}(S)$ , see also [13, 2.10.7] or [14, 6.2.3.14].

The proof of this result is the blueprint for the proof of Theorem 3. However, those proofs employ an intermediate comparison with the Weyl numbers $x_{n}$ , which lie somehow between $h_{n}$ and $c_{n}$ . Despite interesting consequences, this approach requires the multiplicativity of $x_{n}$ , the notion of 2-summing norm, and some technical difficulties. The proof presented here is elementary: It only uses definitions and known properties of the determinant.

Bounds for individual $n$ cannot be deduced from this approach, see also Remark 5, and it remains a long-standing open problem if $d_{bn}(S)\leq c\,n\,b_{n}(S)$ , or even $a_{bn}(S)\leq c\,n\,h_{n}(S)$ , for some $b,c\geq 1$ , see [16, Prob. 5] or [13, 2.10.7].

3. The proofs

Proof of Proposition 2.

We refer to [13, 2.11.9] for the proof that $s_{n}(S)=a_{n}(S)$ for any s-number sequence $(s_{n})$ , and any $S\in\mathcal{L}(H,K)$ for Hilbert spaces $H$ and $K$ . Just note that, for compact $S$ , this follows quite directly from the singular value decomposition. From this and (S3), we obtain

h_{n}(S)\,=\,\sup\Biggl{\{}\frac{s_{n}(BSA)}{\|B\|\|A\|}\colon A\in\mathcal{L}(\ell_{2},X),\;B\in\mathcal{L}(Y,\ell_{2}),\;A,B\neq 0\Biggr{\}}\,\leq\,s_{n}(S)

for any $(s_{n})$ . In addition, by (S2) and (S5), we obtain for any $L$ with $\operatorname{rank}(L)<n$ that $s_{n}(S)\,\leq\,s_{n}(L)+\|S-L\|\,=\,\|S-L\|$ . By taking the infimum over all such $L$ , we see that $s_{n}(S)\leq a_{n}(S)$ . ∎

Proof of Theorem 3.

We first present the proof from [13, 2.10.3] of the following statement: For fixed $\varepsilon>0$ , we can find $x_{1},\dots,x_{n}\in B_{X}$ and $b_{1},\dots,b_{n}\in B_{Y^{\prime}}$ such that $\left\langle Sx_{k},b_{j}\right\rangle=0$ for $j<k$ and $(1+\varepsilon)|\left\langle Sx_{k},b_{k}\right\rangle|>c_{k}(S)$ for $k=1,\ldots,n$ .

For this, we inductively assume that $x_{k},b_{k}$ for $k<n$ are already found, and define

M_{n}\,:=\,\Bigl{\{}x\in X\colon\left\langle Sx,b_{k}\right\rangle=0\text{ for }k<n\Bigr{\}}.

Since $\operatorname{codim}{M_{n}}<n$ , we can choose $x_{n}\in M_{n}\cap B_{X}$ with

(1+\varepsilon)\|Sx_{n}\|\,\geq\,\|SJ_{M_{n}}^{X}\|\,\geq\,c_{n}(S).

By the Hahn-Banach theorem, we choose $b_{n}\in B_{Y^{\prime}}$ with $\left\langle Sx_{n},b_{n}\right\rangle=\|Sx_{n}\|\geq\frac{c_{k}(S)}{1+\varepsilon}$ .

We now define the operators

A(\xi)\,:=\,\sum_{i=1}^{n}\xi_{i}x_{i}\in X,\quad\xi=(\xi_{i})\in\ell_{2}^{n},

and

B(y)\,:=\,\bigl{(}\left\langle y,b_{i}\right\rangle\bigr{)}_{i=1}^{n}\in\ell_{2}^{n},\qquad y\in Y,\vskip 6.0pt plus 2.0pt minus 2.0pt

which satisfy $\|A\|,\|B\|\leq\sqrt{n}$ , and observe that $S_{n}:=BSA\colon\ell_{2}^{n}\to\ell_{2}^{n}$ is generated by the triangular matrix $(\left\langle Sx_{i},b_{j}\right\rangle)_{i,j=1}^{n}$ with determinant $\det(S_{n})\geq\prod_{k=1}^{n}\frac{c_{k}(S)}{1+\varepsilon}$ .

To obtain a bound with s-numbers, note that they all coincide for $S_{n}$ , esp. with $a_{k}(S_{n})$ , and are equal to the singular numbers $s_{k}(S_{n})$ , i.e., the roots of the eigenvalues of $S_{n}S_{n}^{*}$ , see [14, 6.2.1.2]. As the determinant is multiplicative and equals the product of the eigenvalues, we see that $\det(S_{n})=\sqrt{\det(S_{n}S_{n}^{*})}=\prod_{k=1}^{n}a_{k}(S_{n})$ . From the definition of $h_{n}$ , we obtain $a_{k}(S_{n})\leq\|A\|\|B\|\,h_{k}(S)\leq n\cdot h_{k}(S)$ , and hence

\begin{split}(1+\varepsilon)^{-n}\prod_{k=1}^{n}c_{k}(S)\;&\leq\;\det(S_{n})\;=\;\prod_{k=1}^{n}a_{k}(S_{n})\;\leq\;n^{n}\prod_{k=1}^{n}h_{k}(S).\end{split}

With $\varepsilon\to 0$ and $c_{n}(S)\leq\bigl{(}\prod_{k=1}^{n}c_{k}(S)\bigr{)}^{1/n}$ we obtain the result for $c_{n}(S)$ .

The proof for $d_{n}(S)$ could be done via duality, at least for compact $S$ , see e.g. [14, 6.2.3.9 & 6.2.3.12]. However, one can also prove it directly by inductively choosing $M_{n}:=\operatorname{span}\{Sx_{k}\colon k<n\}$ , $x_{n}\in B_{X}$ with $(1+\varepsilon)\|Q^{Y}_{M_{n}}Sx_{n}\|\,\geq\,\|Q^{Y}_{M_{n}}S\|\,\geq\,d_{n}(S)$ , and $b_{n}\in B_{Y^{\prime}}$ with $\left\langle Sx_{n},b_{n}\right\rangle=\|Q^{Y}_{M_{n}}Sx_{n}\|$ and $\left\langle Sx_{k},b_{n}\right\rangle=0$ for $k<n$ . The remaining proof is as above. ∎

Remark 5.

The proof of Theorem 3 uses the determinant to relate the eigenvalues $\lambda_{k}(S_{n})$ of $S_{n}$ (which are the diagonal entries) with its singular numbers. Sometimes, the more general Weyl’s inequality [21] from 1949 is used, which states that $\prod_{k=1}^{n}|\lambda_{k}(S)|\;\leq\;\prod_{k=1}^{n}a_{k}(S)$ for any compact $S\in\mathcal{L}(H,H)$ , see also [14, 3.5.1]. This crucial step appears in all the proofs I am aware of that lead to the optimal factor $n$ in the comparisons. Unfortunately, all these approaches use a whole collection of s-numbers, which does not allow for bounds for individual $n$ .

Let us present an example from [5, 2.d.5] that shows that such product bounds between eigenvalues and s-numbers are to some extent best possible:
For $0<\sigma<1$ , consider the matrix $T_{n}=(\delta_{j,i+1}+\sigma\cdot\delta_{i,n}\delta_{j,1})_{i,j=1}^{n}$ which represents a mapping on $\ell_{2}^{n}$ . It is easy to verify that $a_{k}(T_{n})=1$ for $k<n$ and $a_{n}(T_{n})=\sigma$ . (Recall that $a_{k}$ are the singular numbers in this case.) Moreover, since $T_{n}^{n}=\sigma\cdot I_{\ell_{2}^{n}}$ , we see that $|\lambda_{k}(T_{n})|=\sigma^{1/n}$ for $k=1,\dots,n$ . This shows that Weyl’s inequality, as well as the easy corollary $|\lambda_{n}(S)|\leq\|S\|^{1-\frac{1}{n}}a_{n}(S)^{1/n}$ , are in general best possible.

Acknowledgement: I thank Albrecht Pietsch for comments on an earlier version, and for encouraging me to make this note as self-contained as it is now. The last comments I received from him were on March 8, 2024, where he added that he was busy with his own article. My reply, however, was answered by his granddaughter, who informed me that Professor Pietsch had passed away on March 10.
I also thank S. Heinrich, A. Hinrichs, D. Krieg, T. Kühn, E. Novak and the anonymous referees for helpful comments.

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