Greedy Algorithms and Kolmogorov Widths in Banach Spaces

Greedy Algorithms and Kolmogorov Widths in Banach Spaces Van Kien Nguyen∗

arXiv:1804.03935v1 [math.FA] 11 Apr 2018

April 12, 2018

Abstract Let X be a Banach space and K be a compact subset in X. We consider a greedy algorithm for finding n-dimensional subspace Vn ⊂ X which can be used to approximate the elements of K. We are interested in how well the space Vn approximates the elements of K. For this purpose we compare the greedy algorithm with the Kolmogorov, width which is the best possible error one can approximate K by n−dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (SIAM J. Math. Anal. (2011)), DeVore et al. (Constr. Approx. (2013)) and Wojtaszczyk (J. Math. Anal. Appl. (2015)). The purpose of the present paper is to continue this line. We shall show that under some additional assumptions the results in the above-mentioned papers can be improved.

1

Introduction

Recently, a new greedy algorithm for obtaining a good subspace Vn of n-dimension to approximate elements of a compact set K in a Banach space X has been given. This greedy algorithm was studied initially when X is a Hilbert space in the context of reduced basis methods for solving families of PDEs, see [6, 7]. Later, it was studied extensively not only in the setting of Hilbert spaces, let us mention, for instance, Binev et al. [1], Buffa et al. [2], DeVore et al. [3], and Wojtaszczyk [13]. The greedy algorithm for generating the subspace Vn to approximate elements of K is implemented as follows. We first select f0 such that kf0 kX = max kf kX . f ∈K

Since K is compact, such a f0 always exists. At the general step, assuming that {f0 , . . . , fn−1 } and Vn = span{f0 , . . . , fn−1 } have been chosen, then we take fn such that dist(fn , Vn )X = max dist(f, Vn )X . f ∈K

The error in approximating the elements of K by Vn is defined as σ0 (K)X := kf0 kX ,

σn (K)X := dist(fn , Vn )X = max dist(f, Vn )X f ∈K

for n ≥ 1. The sequence σn (K)X is non-increasing. It is important to note that the sequence {fn }n≥0 and also σn (K)X are not unique. Let us mention that the best possible error one can achieve to approximate the elements of K by n-dimensional subspaces is the Kolmogorov width dn (K)X , which is given by dn (K)X := inf sup dist(f, L)X , L f ∈K

∗

E-mail: [email protected], [email protected]

1

n ≥ 1,

where the infimum is taken over all n-dimensional subspaces L of X. We also put d0 (K)X = max kf kX . f ∈K

We would like to emphasize that in practice, finding subspaces which give this performance is out of reach. We are interested in how well the subspaces created by the greedy algorithm approximate the elements of K. For this purpose it is natural to compare σn (K)X with the Kolmogorov width dn (K)X . Various comparisons between σn (K)X and dn (K)X have been made. The first attempt in this direction was given in [2] and improved in [1], where the authors considered the case when X is a Hilbert space H. Under this assumption, it has been shown that σn (K)H ≤ C2n dn (K)H for an absolute constant C. Observe that this result is useful only when dn (K)H decays faster than 2−n . A significant improvement of the above result was given in [3] where the authors prove that if the Kolmogorov width has polynomial decay with rate n−s , then the greedy algorithm also yields the same rate, i.e., σn (K)H ≤ Cn−s . In the same paper, the estimate of this type for 1 Banach spaces X was also considered, but there is an additional factor n 2 +ε (for any ε > 0), that is, 1 (1.1) σn (K)X ≤ Cn−s+ 2 +ǫ

where C depends on s and ε. For a recent result in this direction we refer to [13]. Let γ˜n (X) be the supremum of BanachMazur distance d(Vn , ℓn2 ) where Vn is the n-dimensional space in a quotient space of X. If dn (K)X ≤ C0 n−s and γ˜n (X) ≤ C1 nµ , then Wojtaszczyk [13] shows that there is a constant C such that   log(n + 2) s µ n . (1.2) σn (K)X ≤ C n √ Observe that the estimate given in (1.2) improves the result (1.1) since γ˜n (X) ≤ n. It has been shown in [13] that the above estimate is optimal in Lp up to a logarithmic factor. However, for a given Banach space X, the factor γ˜n (X) is not easy to compute. Hence, this raises the question whether we can replace the condition on γ˜ (X) by γn (X) = supVn d(Vn , ℓn2 ) where the supremum is taken over n-dimensional subspaces Vn in X, see Section 2 for the definition. In the present paper we will give a new analysis of the performance of the greedy algorithm in which we show that the assumption on γ˜n (X) can be relaxed to γn (X). In addition the rate of the logarithm in (1.2) can also be improved when s > 1/2. More precisely we shall prove that there is a constant C > 0 such that p σn (K)X ≤ C log(2n) n−s+µ

if dn (K)X ≤ C0 n−s and γn (X) ≤ C1 nµ . Often, the compact set of interest K is the image (or subset) of the closed unit ball BE of a Banach space E under a compact operator T ∈ L(E, X). For this reason, we shall compare σn (K)X with the Kolmogorov widths dn (T (BE ))X . In this study, we obtain the estimate  n−1 1/n Y 2 σ3n−1 (K)X ≤ 3e Γ(E)Γn (X) dk (T (BE ))X , n ≥ 1, (1.3) k=0

here Γn (X) is the n-Grothendieck number of X, which is closely related to γn (X), see Section 2. Note that if E is a Hilbert space then Γ(E) = 1. In Section 3 we will give an example showing that the estimate (1.3) is sharp in some situations. The rest of our paper is organized as follows. In the next Section 2 we will collect some required tools. The main results are stated and proved in Section 3. 2

2

Some preparations

In this section we collect some tools needed to formulate our results in the next section. The Banach - Mazur distance of two isomorphic Banach spaces X and Y is defined by  d(X, Y ) = inf kT k · kT −1 k : T : X → Y is an isomorphism . For a Banach space X we introduce a sequence of numbers  γn (X) = sup d(V, ℓn2 ) , V is an n-dimensional subspace in X .

The sequence γn (X) is non-decreasing and γ1 (X) = 1. It is obvious that if X is a Hilbert space then we have γn (X) = 1, n = 1, 2, 3, . . .. In the case of an arbitrary Banach space X, it is |1−1| known that γn (X) ≤ n1/2 and γn (Lp ) ≤ n 2 p for 1 ≤ p ≤ ∞. Let X and Y be Banach spaces of finite dimension. Then there exists an operator T : X → Y such that d(X, Y ) = kT k · kT −1 k. We can additionally assume that kT −1 k = 1. Hence a new norm on X defined by kxke := kT xkY satisfies kxkX ≤ kxke ≤ d(X, Y )kxkX .

(2.1)

Moreover T is an isometry between (X, k · ke ) and Y . The local injective distance γn (X) is closely related to the so-called Grothendieck number. Let T ∈ L(X, Y ) be a linear bounded operator. The n-th Grothendieck number of T is defined as n o
1/n Γn (T ) := sup det hT xi , bj i , x1 , . . . , xn ∈ BX , b1 , . . . , bn ∈ BY ′ . If T is the identity map of X then we write Γn (X). Let 0 ≤ δ ≤ 1/2. A Banach space X is said to be of weak Hilbert type δ if there exists a constant C ≥ 1 such that Γn (X) ≤ C nδ for n ≥ 1. We denote the class of these spaces by Γδ . Note that Γ1/2 is the set of all Banach spaces, i.e., Γn (X) ≤ cn1/2 ,

for all X .

In particular we have Γn (Lp ) ≤ n|1/p−1/2| for 1 ≤ p ≤ ∞. The relation between γn (X) and Grothendieck numbers is represented in the following lemma, see, e.g., [10]. Lemma 2.1. Let X be a Banach space. Then γn (X) ≤ C1 nδ

Γn (X) ≤ C2 nδ

if and only if

for some C1 , C2 ≥ 1. For later use, let us introduce the notion of Kolmogorov and Gelfand widths of linear continuous operators. In the following we use the definition given in [12, Chapter 2], but see also [8, Chapter 11]. Note that there is a shift of 1 between definitions by Pinkus [12, Chapter 2] and Pietsch [8, Chapter 11]. Let BX be the closed unit ball of X. The Kolmogorov n-width of the operator T ∈ L(X, Y ) is defined as dn (T ) := dn (T (BX ))Y = inf sup

inf kT x − ykY ,

Ln kxkX ≤1 y∈Ln

where the infimum is taken over all subspace Ln of dimension n in Y . The Gelfand n-th width of T ∈ L(X, Y ) is given by dn (T ) := dn (T (BX ))Y := inf n L

3

sup kxkX ≤1 ,x∈Ln

kT xkY ,

where the infimum is taken over subspaces Ln of X of co-dimension at most n. We also put d0 (T ) = d0 (T ) = kT k. Note that Kolmogorov and Gelfand widths are closely related, i.e., dn (T ) = dn (T ′ ) for every T ∈ L(X, Y ) and dn (T ) = dn (T ′ ) if T is compact or Y is a reflexive Banach space, see [12, Chapter 2]. Here recall that T ′ is the dual operator of T . For basic properties of these quantities we refer to monographs [12, Chapters 2] and [8, Chapter 11] . The relation between Grothendieck number and Kolmogorov, Gelfand widths is given in the following lemma. For a proof we refer to [10]. Lemma 2.2. Let X and Y be Banach spaces and T ∈ L(X, Y ). Then it holds  n−1 Y k=0

1/n ≤ Γn (T ) dk (T )

and

 n−1 Y k=0

1/n ≤ Γn (T ) d (T ) k

for all n ≥ 1. An operator T ∈ L(X, Y ) is called absolutely 2-summing if there exists a constant C such that   X 1/2 X 1/2 n n 2 ′ 2 ′ |hxi , bi| (2.2) kT xi k : b ∈ X , kbkX ≤ 1 . ≤ C sup i=1

i=1

The set of these operators is denoted by B2 (X, Y ) and the norm kT |B2 k is given by the infimum of all C > 0 satisfying (2.2). The following assertion can be found in [10]. Lemma 2.3. Let X and Y be Banach spaces. Let T ∈ B2 (X, Y ). Then we have Γn (T ) ≤ en−1/2 kT |B2 k Γn (X) ,

3

n ≥ 1.

Main results

Our first result can be formulated as follows. Theorem 3.1. Let X be a Banach space and K a compact subset of X. Assume that dn (K)X ≤ C0 max(1, n)−s , (n ≥ 0) for 0 ≤ µ ≤

1 2

and

γn (X) ≤ C1 nµ , (n ≥ 1)

and s > µ. Then we have σn (K)X ≤ C0 C1 2µ 16s

p

log(2n) n−s+µ

for n ≥ 2 .

(3.1)

Proof. The idea of the proof follows from the proof of Proposition 2.2 in [13]. Step 1. Let ε > 0. From the assumption dn (K)X ≤ C0 n−s , n ≥ 1, we infer the existence of a sequence of subspaces (Tk )k≥0 in X and dim(Tk ) = 2k such that max min kx − gkX ≤ (C0 + ε)2−sk . x∈K g∈Tk

For n ∈ N fixed we put Vk = T0 + T1 + . . . + Tk−1 for k = 1, . . . , n. Then we have Vk ⊂ Vk+1 and dim(Vk ) < 2k . Observe that max min kx − gkX ≤ max min kx − gkX ≤ (C0 + ε)2−s(k−1) . x∈K g∈Vk

x∈K g∈Tk−1

(3.2)

We denote N = 2n . Implementing the greedy algorithm for the set K we get the sequence {f0 , . . . , fN −1 }. Then it follows from (3.2) that kfℓ − gℓk kX ≤ (C0 + ε)2−s(k−1) ,

ℓ = 0, . . . , N − 1 ; k = 1, . . . , n 4

(3.3)

for some gℓk ∈ Vk . Let X = span{f0 , . . . , fN −1 } and Y = span{Vn , X}. It is obvious that 2n ≤ dim(Y ) < 2n+1 . From (2.1) we infer the existence of a Euclidean norm k · ke on Y satisfying dim(Y )
kykX ≤ kyke ≤ d Y, ℓ2 kykX ≤ γdim(Y ) (X)kykX ≤ AkykX , (3.4)

where we put A = γ2n+1 (X). Let Q be the orthogonal projection from Y onto X in the Euclidean norm k · ke . We denote dim(Q(Vk )) = hk for k = 1, . . . , n. It is clear that hk ≤ dim(Vk ) < 2k and Q(Vk−1 ) ⊂ Q(Vk ). From (3.3) and (3.4) we get dist(fℓ , Q(Vk ))k·ke ≤ kfℓ − Q(gℓk )ke

= kQ(fℓ − gℓk )ke ≤ kfℓ − gℓk ke ≤ (C0 + ε)A2−s(k−1) .

(3.5)

By {φj }j=0,...,N −1 we denote the orthonormal system obtained from f0 , . . . , fN −1 by GramN −1 Schmidt orthogonalization in the norm k · ke . It follows that the matrix [φj (fℓ )]j,ℓ=0 has a triangular form. In particular, on the diagonal we have

(3.6) dist fℓ , span{f0 , . . . , fℓ−1 } k·ke ≥ dist fℓ , span{f0 , . . . , fℓ−1 } X = σℓ (K)X . Step 2. We consider the case 0 < hm1 = . . . = hm2 −1 < hm2 = . . . = hm3 −1 < . . . < hmL = . . . = hn where m1 = 1, mL+1 = n + 1. We denote {xj }j=0,...,N −1 another orthonormal basis in X, such that Q(Vmi−1 ) = . . . = Q(Vmi −1 ) = span{x0 , . . . , xhmi−1 −1 } N −1 for i = 2, . . . , L. Considering the vector [xj (fℓ )]j=0 we observe that N −1 X

j=hmL

and

|xj (fℓ )|2 = dist(fℓ ; Q(Vn ))2k·ke

(3.7)

hmi −1 2

|x0 (fℓ )| ≤

kfℓ k2e ,

X

j=hmi−1

|xj (fℓ )|2 ≤ dist(fℓ ; Q(Vmi −1 ))2k·ke ,

(3.8)

for i = 2, . . . , L. Note that N −1 Y j=0

σj (K)X ≤

N −1 Y j=0

|φj (fj )| = det[φj (fℓ )] = det[xj (fℓ )] ,

N −1 . Applying Hadamard’s see (3.6). By kj we denote the j-th column of the matrix [xj (fℓ )]j,ℓ=0

5

inequality and then arithmetic-geometric mean inequality we obtain  NY −1

σj (K)X

j=0

2


2 ≤ det[xj (fℓ )] ≤

 hY 1 −1

≤



j=0

kkj k2e

 NY −1

j=hmL

h1 −1 1 X kkj k2e h1

×

j=0

L  Y i=2

hmi

h 1 

kkj k2e

 Y L

hmi −1

Y

i=2 j=hmi−1

kkj k2e



N −hm N −1 X L 1 2 kkj ke N − hmL

(3.9)

j=hmL

1 − hmi−1

hmi −1

X

j=hmi−1

kkj k2e

i −hmi−1

h m

.

From (3.5), (3.7), and (3.8) we have hX 1 −1 j=0

kkj k2e =

hX −1 1 −1 N X j=0 ℓ=0

|xj (fℓ )|2 ≤

N −1 X ℓ=0

kfℓ k2e ≤ N A2 d0 (K)2X ≤ N A2 (C0 + ε)2

(since d0 (K)X ≤ C0 , by our assumption), and for i = 2, . . . , L, hmi −1 N −1 X X

hmi −1

X

j=hmi−1

kkj k2e

=

j=hmi−1 ℓ=0

≤

N −1 X ℓ=0

|xj (fℓ )|2

dist(fℓ ; Q(Vmi −1 ))2k·ke ≤ N (C0 + ε)2 A2 2−2s(mi −2) .

Similarly, we have N −1 X

j=hmL

kkj k2e ≤

N −1 X ℓ=0

dist(fℓ ; Q(Vn ))2k·ke ≤ N (C0 + ε)2 A2 2−2s(n−1) .

Inserting this into (3.9) we find  NY −1

σj (K)X

j=0

2

≤



N A2 (C0 + ε)2 h1 ×

h 1 

N (C0 + ε)2 A2 2−2s(n−1) N − hmL

N −hm

L

 L  Y N (C0 + ε)2 A2 2−2s(mi −2) hmi −hmi−1

hmi − hmi−1   L Y −2s(mi −2)(hmi −hmi−1 ) 2N 2N −2s(n−1)(N −hmL ) = M A (C0 + ε) 2 2 , i=2

i=2

where we put M=



N h1

h 1 

N N − hmL

N −hm Y L  L i=2

6

hmi

N − hmi−1

i −hmi−1

hm

.

(3.10)

For nonnegative number a1 , . . . , an and positive numbers p1 , . . . , pn we have   a1 p1 + . . . + an pn p1 +...+pn p1 pn , a1 · · · an ≤ p1 + . . . + pn see, e.g., [4, Page 17]. Applying the above inequality for M we get M ≤ (L + 1)N ≤ (n + 1)N .

(3.11)

Now we deal with the term   L Y −2s(mi −2)(hmi −hmi−1 ) −2s(n−1)(N −hmL ) 2 U := 2

i=2 2s[−(n−1)2n +(n−1)hmL −(mL −2)(hmL −hmL−1 )−...−(m2 −2)(hm2 −hm1 )]

=2

n +h

= 22s[−(n−1)2

mL (n+1−mL )+hmL−1 (mL −mL−1 )+...+hm2 (m3 −m2 )+hm1 (m2 −2)]

.

Using hmi < 2mi for i = 1, . . . , L we can estimate n +2n +...+22 +21 ]

U ≤ 22s[−(n−1)2

n

≤ 22s[−(n−3)2 ] .

(3.12)

Plugging (3.11) and (3.12) into (3.10) we obtain  NY −1 j=0

σj (K)X

2

n ]2s

= (n + 1)N A2N (C0 + ε)2N 2[(−n+3)2

.

Finally from the assumption A ≤ C1 2µ(n+1) we find √ σ2n −1 (K)X ≤ (C0 + ε)C1 n + 1 · 2(n+1)µ 2(−n+3)s √ = (C0 + ε)C1 8s n + 1 · 2(n+1)µ 2−ns and hence 2n

2n+1 .

σj (K)X ≤ (C0 + ε)C1 2µ 16s

p

log(2j) · j (µ−s)

(3.13)

for ≤j< Since ε > 0 arbitrary we get (3.1). Step 3. We comment on the case

0 = hm1 = . . . = hm2 −1 < hm2 = . . . = hm3 −1 < . . . < hmL = . . . = hn . In this situation we proceed as in Step 2, but there is no first term in the product on the righthand side of (3.9). Note that in case h1 = . . . = hn = 0, there is no logarithmic factor on the right-hand side of (3.13). The proof is complete. Remark 3.2. Comparing with Theorem 2.3 in [13] we found that in the case s > 1/2 the estimate given in Theorem 3.1 improves the rate of the logarithmic term. Moreover the factor γ˜n (X) is replaced by γn (X), which is somewhat better, since in general γn (X) ≤ γ˜n (X) . In the case of Lebesgue spaces we have the following. Corollary 3.3. Let 1 ≤ p ≤ ∞, s > 21 − p1 , and K be a compact set in Lp . Assume that dn (K)Lp ≤ C0 max(1, n)−s , n ≥ 0 .

Then we have | 12 − p1 |

σn (K)Lp ≤ C0 16s 2

p

log(2n) n 7

−s+| 21 − p1 |

for n ≥ 2 .

Let E and X be Banach spaces and BE be the closed unit ball of E. As a supplement we study the case K ⊂ T (BE ) where T ∈ L(E, X) is a compact operator. We shall compare the rate of convergence of σn (K) with the Kolmogorov widths of T (BE ). In this situation we have the following. Theorem 3.4. Let X be a Banach space and K be a compact set in X. Assume that there exists a compact operator T ∈ L(E, X) where E is a reflexive Banach space such that K ⊂ T (BE ). Then we have 1/n  n−1  3n−1 1/3n Y Y 2 , n ≥ 1. dk (T ) σk (K)X ≤ 3e Γn (E)Γn (X) k=0

k=0

Proof. First, note that T (BE ) is a closed set in X since T is a compact operator and E is reflexive. For n ∈ N fixed, running the greedy algorithm for K we get {f0 , . . . , f3n−1 } and Vk = span{f0 , . . . , fk−1 }. We select ek ∈ BE such that T ek = fk for k = 0, . . . , 3n − 1. For each k ∈ N, as a consequence of the Hahn-Banach Theorem, see [5, Corollary 14.13], we can choose bk ∈ X ′ such that kbk kX ′ = 1, hVk , bk i = 0 ,

and

hfk , bk i = hT ek , bk i = dist(fk , Vk )X = σk (K)X .

(3.14)

3n We define the operators A ∈ L(ℓ3n 2 , E) and B ∈ L(X, ℓ2 ) by

A :=

3n−1 X k=0

uk ⊗ ek

and

B :=

3n−1 X k=0

bk ⊗ uk ,

(3.15)

ℓ3n 2 .

We calculate the norm k B |B2 k, see the where {uk }k=0,...,3n−1 is the canonical basis of definition (2.2). Let x1 , . . . , xN ∈ X. We have 1/2 X 1/2  X N 3n−1 N 3n−1 N

2 1/2  X X

X

2 2 |hxi , bk i| = hxi , bk iuk 3n kBxi kℓ3n =

2

i=1

i=1

ℓ2

k=0

i=1 k=0

which implies

X N i=1

kBxi k2ℓ2

1/2

≤ ≤

√

3n

sup k=0,...,3n−1

√

3n sup b∈BX ′

X N

X N i=1

i=1

2

|hxi , bk i| 2

|hxi , bi|

1/2

1/2

.

√ Hence k B |B2 k ≤ 3n . We consider the matrix (hT ek , bj i) = (hBT Auk , uj i) which has the lower triangular form. It follows from (3.14) that  3n−1 1/3n 1/3n  3n−1 Y Y
1/3n σk (K)X . = det hBT Aui , uj i = |hT ek , bk i| k=0

k=0

Qn−1 dk (S) since Kolmogorov widths Note that for any operator S ∈ L(ℓn2 ) we have | det S| ≤ k=0 equal to singular values of S, see [9]. Consequently we obtain 1/3n  3n−1 1/3n  3n−1 Y Y dk (BT A) σk (K)X ≤ k=0

k=0

=

 n−1 Y k=0

d3k (BT A)

n−1 Y k=0

8

d3k+1 (BT A)

n−1 Y k=0

1/3n . d3k+2 (BT A)

(3.16)

From the property dm+n+k (BT A) ≤ dm (B)dn (T )dk (A), see [12, Page 32] or [8, Theorem 11.9.2], and the monotonicity dk+1 ≤ dk of Kolmogorov widths we conclude that  3n−1 1/n  n−1 1/n  n−1 1/3n  n−1 1/n Y Y Y Y σk (K)X ≤ . (3.17) dk (A) dk (T ) dk (B) k=0

k=0

k=0

k=0

Lemmas 2.2 and 2.3 yield the estimate  n−1 1/n Y ≤ Γn (B) ≤ en−1/2 kB|B2 k Γn (X) dk (B) k=0

≤ en

−1/2

1/2

(3n)

√ Γn (X) = e 3 Γn (X) .

(3.18)

Now we deal with the first product on the right-hand side of (3.17). We have  n−1 1/n  n−1 1/n Y Y k ′ = ≤ Γn (A′ ) , dk (A) d (A ) k=0

k=0

the form A′ = see Section 2. Here A′ ∈ L(E ′ , ℓ3n 2 ) is the dual operator of A which is of √ P 3n−1 ′ 3n. Hence we k=0 ek ⊗ uk . Similar argument as for the operator B we also get kA |B2 k ≤ found √ Γn (A′ ) ≤ en−1/2 kA′ |B2 k Γn (E ′ ) ≤ en−1/2 (3n)1/2 Γn (E ′ ) = e 3 .Γn (E ′ ) which leads to  n−1 Y k=0

1/n √ √ ≤ e 3 Γn (E ′ ) = e 3 Γn (E) . dk (A)

Putting this and (3.18) into (3.17) we arrive at  n−1  3n−1 1/3n 1/n Y Y 2 ≤ 3e Γn (E)Γn (X) . dk (T ) σk (K)X k=0

k=0

The proof is complete. Remark 3.5. Note that Theorem 3.4 still holds true if one replaces Kolmogorov widths by Gelfand widths, i.e.,  3n−1  n−1 1/n 1/3n Y Y 2 k σk (K)X , n ≥ 1. ≤ 3e Γn (E)Γn (X) d (T ) k=0

k=0

We have the following consequence. Corollary 3.6. Let X be a Banach space and K be a compact set in X. Assume that there exists a compact operator T ∈ L(ℓ2 , X) such that K ⊂ T (Bℓ2 ). Then we have  n−1 1/n Y 2 σ3n−1 (K)X ≤ 3e Γn (X) , n ≥ 1. (3.19) dk (T ) k=0

In addition, if X = Lp for 1 ≤ p ≤ ∞ and dn (T ) ≤ C0 n−s for some s > 12 − 1p , (n ≥ 1), then there exists a constant C > 0 such that σ3n−1 (K)X ≤ C3e2 n|1/p−1/2|−s , 9

n ≥ 1.

We proceed by considering the case K = T (Bℓ2 ) for some compact operator T ∈ L(ℓ2 , X). In this situation we can replace σ3n−1 (K)X in (3.19) by σ2n−1 (K)X . We have the following. Theorem 3.7. Let X be a Banach space and T ∈ L(ℓ2 , X) be a compact operator. Assume that K = T (Bℓ2 ). Then we have σ2n−1 (K)X

 n−1 1/n Y √ dk (K)X , ≤ e 2Γn (X)

n ≥ 1.

k=0

Proof. Recall that T (Bℓ2 ) is the closed set in X. For n ∈ N fixed, running the greedy algorithm for K we get {f0 , . . . , f2n−1 } and Vk = span{f0 , . . . , fk−1 }. First we show that we can select 2n−1 ek ∈ Bℓ2 such that T ek = fk for k = 0, . . . , 2n − 1 and {ek }k=0 is an orthonormal system in ℓ2 . Indeed, if T e0 = f0 with kf0 kX = max kf kX = max kT ekX , e∈Bℓ2

f ∈K

then ke0 kℓ2 = 1. Assume that we have chosen the orthonormal system {e0 , . . . , ek−1 } in ℓ2 with T ei = fi for i = 0, . . . , k − 1. Let {ej }j≥0 be an orthonormal basis of ℓ2 constructed from the P system {e0 , . . . , ek−1 } and let fk = T e where e = j≥0 cj ej with k(cj )j≥0 kℓ2 ≤ 1. We consider fk∗ =

1 X cj T ej , kc∗ kℓ2

with c∗ = (0, . . . , 0, ck , ck+1 , . . .) .

j≥k

Here we assume that c∗ 6= 0 otherwise σk (K)X = 0. We have fk∗ ∈ T (Bℓ2 ) and dist(fk , Vk ) ≥ dist(fk∗ , Vk )X

k−1 X

1 X

= inf ∗ aj T ej cj T ej −

a0 ,...,ak−1 kc kℓ2 X j=0

j≥k

= =

1 kc∗ kℓ2 1 kc∗ kℓ2

X k−1 X


∗

aj kc kℓ2 + cj T ej cj T ej − inf

a0 ,...,ak−1 j=0

j≥0

X

dist(fk , Vk ) .

Hence kc∗ kℓ2 = 1 and fk = fk∗ which implies e is orthogonal to {e0 , . . . , ek−1 } and kekℓ2 = 1. 2n Similar to (3.15) we define the operators A ∈ L(ℓ2n 2 , ℓ2 ) and B ∈ L(X, ℓ2 ) by A :=

2n−1 X k=0

uk ⊗ ek

and

B :=

2n−1 X k=0

bk ⊗ u k .

√ Note that kB|B2 k ≤ 2n and kAk ≤ 1 . We have  2n−1 Y k=0

σk (K)X

1/2n

≤

 2n−1 Y

1/2n dk (BT A)

≤

 2n−1 Y

1/2n 1/2n  2n−1 Y , ≤ dk (BT ) kAkdk (BT )

k=0

k=0

k=0

see (3.16). By the same argument as in the proof of Theorem 3.4 we obtain the desired estimate.

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In some situations, the estimate given in Corollary 3.6 is sharp. Let us consider the following example which is borrowed from [3], see also [13]. Let K = {n−α un } ⊂ ℓq with 2 < q < ∞ 1 where {un }n≥1 is the canonical basis of ℓ2 . It is clear that σn (K)ℓq = (n+1) α . We consider the −α diagonal operator Dα : ℓ2 → ℓq defined by un → n un . Then K ⊂ Dα (Bℓ2 ). We know that 1

1

dn (Dα ) ≤ Cn−α+ q − 2 , see, e.g., [11, Section 6.2.5.3] which implies σ3n−1 (K)ℓq ≤ Cn

| 21 − 1q | −α+ q1 − 21

n

= Cn−α .

Hence the operator Dα give the sharp estimate in the rate of convergence in this example. Acknowledgements The author would like to thank Markus Bachmayr for fruitful discussions. Moreover, the author acknowledges the Hausdorff Center of Mathematics, University of Bonn for financial support.

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