Georgian Mathematical Journal Volume 8 (2001), Number 2, 307–318

ENTROPY NUMBERS OF DIAGONAL OPERATORS OF LOGARITHMIC TYPE

THOMAS KÜHN

Dedicated to Professor Nicholas Vakhania on his 70th birthday

Abstract. We determine the asymptotic behaviour (as k → ∞, up to multiplicative constants not depending on k) of the entropy numbers e_{k} (D_{σ} : l_{p} → l_{q}), 1 ≤ p ≤ q ≤ ∞, of diagonal operators generated by logarithmically decreasing sequences σ = (σ_{n}). This complements earlier results by Carl [2] who investigated the case of power-like decay of the diagonal.

2000 Mathematics Subject Classification: 47B06, 46B45, 46B70, 28C20. Key words and phrases: Entropy numbers, sequence spaces, complex interpolation, Gaussian processes.

1. Introduction

In the last few years there has appeared quite a number of research papers (see [17], [18], [7] and [8]) and several monographs (see [9] and [25]) on compact embeddings of function spaces in so-called limiting situations, where the spaces have the same ‘differential dimension’. The ‘degree’ of compactness of a (bounded linear) operator can be measured by the rate of decay of its entropy numbers. Standard tools in the treatment of such embedding problems are discretization techniques, which allow to reduce an infinite dimensional problem to a finite dimensional one, or which reduce function space embeddings to embeddings of weighted vector-valued sequence spaces. This approach requires sharp estimates for the entropy numbers of the latter embeddings. Equivalently, by shifting the weights from spaces to operators, one can consider diagonal operators in sequence spaces without weights.

For classical non-limiting embeddings these weights are of power-like decay. In the above-mentioned limiting situations, however, the resulting weights decrease only logarithmically. This was one of the motivations for the present work.

A further motivation came from the paper by Carl [2] who determined the entropy behaviour of diagonal operators D_{σ} : l_{p} → l_{q}, 1 ≤ p, q ≤ ∞, in the case where the diagonal σ belongs to some Lorentz space l_{r,t}. Typically, for

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

t = ∞, this means σ_{n} ⪯ n^{−α} with α = 1/r. We give in this paper sharp entropy estimates for diagonal operators D_{σ} in the following two cases:

(P) σ_{n} ∼ n^{−α}(1 + log n)^{δ}, 1 ≤ p, q ≤ ∞, α > max(1/q − 1/p, 0), δ ∈ ℝ; (L) σ_{n} ∼ (1 + log n)^{−δ}, 1 ≤ p ≤ q ≤ ∞, δ > 0. The first case (P), of power-like decay of the diagonal, is a variant of Carl’s

result [2], and the proof also relies on the same, by now standard, technique of splitting the operator D_{σ} in dyadic parts. The logarithmic case (L), however, extends the results of [2] to the limiting situation r = ∞ or, equivalently, α = 0. Here the dyadic splitting does not give satisfactory estimates and so a new technique is needed.

Our new approach (recently introduced in two joint papers of the author with Cobos [5] and Schonbek [15]) is essentially based on a close relation between Gaussian processes (resp. Gaussian measures) and entropy. More precisely, we use Sudakov’s famous minoration principle – in the form of an inequality between entropy numbers and the l-norm of an operator – in combination with certain interpolation techniques.

The organization of the paper is as follows. In Section 2 we fix the notation, and give a few preliminary results which will be needed in the sequel. In Section 3 the main results are formulated, while Section 4 contains their proofs.

2. Notation and Preliminary Results

a. Entropy numbers and ideals. The k^{th} (dyadic) entropy number of a bounded linear operator T : X → Y from a Banach space X into another Banach space Y is defined as

e_{k}(T) = inf{ε > 0 : T (B_{X}) can be covered by 2^{k−1} balls of radius ε in Y },

where B_{X} denotes the closed unit ball in X. Entropy numbers are closely related to Kolmogorov’s concept of metric entropy which dates back to the 1930’s and which since then has been successfully applied in various fields of mathematics. For basic properties of entropy numbers and applications to eigenvalue and compactness problems we refer to the monographs by Pietsch [21], König [12], Carl and Stephani [4], Edmunds and Triebel [9], and Triebel [25], and the references given therein; for recent applications see also Edmunds and Haroske [7, 8], and Leopold [17, 18].

We use the common notation l_{p}^{n} for the space ℝ^{n} (or ℂ^{n}) equipped with the quasi-norm (which is even a norm for p ≥ 1)

∥x∥_{p} = \{ \begin{matrix}( ∑_{k=1}^{n} |x_{k}|^{p})^{1/p} & for & 0 < p < ∞ \\ sup_{1≤k≤n} |x_{k}| & for & p = ∞. \end{matrix}

Given two sequences (a_{k})_{k=1}^{∞} and (b_{k})_{k=1}^{∞} of positive real numbers, we shall write a_{k} ⪯ b_{k} if there is a constant c > 0 such that the inequality a_{k} ≤ c · b_{k} holds for all k ∈ ℕ, and a_{k} ∼ b_{k} if both a_{k} ⪯ b_{k} and b_{k} ⪯ a_{k} are fulfilled. Logarithms are always taken to base 2, log = log_{2}.

The following basic result is due to Schütt [22] who determined the exact asymptotic behaviour of entropy numbers of the formal identity between finite dimensional l_{p}-spaces.

Lemma 1. Let 1 ≤ p ≤ q ≤ ∞, then (in the real case)

e_{k} (id : l_{p}^{n} → l_{q}^{n}) ∼ \{ \begin{matrix}1 & if & 1 ≤ k ≤ log n \\ (\frac{log(\frac{n}{k} +1)}{k})1/p−1/q & if & log n ≤ k ≤ n \\ 2^{− \frac{k−1}{n}} n^{1/q−1/p} & if & k ≥ n. \end{matrix}  (1)

Remark 1. Formula (1) remains valid for complex spaces if on the right-hand side n is replaced by 2n. Roughly speaking, this follows from volume arguments, regarding the n-dimensional complex spaces l_{p}^{n} as 2n-dimensional real spaces.

Remark 2. A consequence of the special case k = n and elementary properties of entropy numbers is the relation

e_{n} (id : l_{p}^{n} → l_{q}^{n}) ∼ n^{1/q−1/p} for all 1 ≤ p, q ≤ ∞.  (2)

The asymptotic relations (1) are fundamental for numerous applications, as explained in the Introduction. In passing, we mention that (1) has recently been extended to the quasi-Banach case 0 < p ≤ q ≤ ∞, see [9, 25, 14]. This was again motivated by applications, mainly by the increasing interest during the last decade in quasi-Banach function spaces, for instance in (both classical and generalized) Besov, Sobolev, and Lipschitz spaces.

Another very useful result is due to Gordon, König and Schütt [11, Proposition 1.7.].

Lemma 2. Let X be a real Banach space with 1-unconditional basis {x_{j}}_{j=1}^{∞} (for example a symmetric Banach sequence space, or any of the spaces l_{p} with 1 ≤ p < ∞), and let D_{σ} : X → X be the diagonal operator induced by x_{j} ↦ σ_{j}x_{j}, where σ_{1} ≥ σ_{2} ≥ ⋯ ≥ 0. Then

sup_{n∈ℕ}2^{−k/n}(σ_{1} ⋯ σ_{n})^{1/n} ≤ e_{k+1}(D_{σ}) ≤ 6 · sup_{n∈ℕ}2^{−k/n}(σ_{1} ⋯ σ_{n})^{1/n}  (3)

for all k ∈ ℕ.

Remark 3. Inequality (3) remains valid for complex spaces if the factor 2^{−k/n} is replaced by 2^{−k/2n}.

As usual, we denote by l_{p,q}, 0 < p < ∞, 0 < q ≤ ∞, the Lorentz sequence space consisting of all (real or complex) null sequences (x_{k})_{k=1}^{∞} such that the quasi-norm

∥x∥_{p,q} := \{ \begin{matrix}( ∑_{k=1}^{∞} k^{q/p−1} |x_{k}^{∗}|^{q})^{1/q} & if & 0 < q < ∞ \\ sup_{1≤k<∞} k^{1/p} |x_{k}^{∗}| & if & q = ∞ \end{matrix}

is finite. Here the non-increasing rearrangement (x_{k}^{∗}) of the sequence (x_{k}) is defined by x_{k}^{∗} = x_{π(k)}, where π : ℕ → ℕ is a permutation with |x_{π(1)}| ≥ |x_{π(2)}| ≥ ⋯ ≥ 0. All Lorentz spaces are quasi-Banach spaces, and for 1 < p < ∞, 1 ≤ q ≤ ∞ and p = q = 1 the quasi-norms ∥·∥_{p,q} are even equivalent to norms. In particular we have l_{p,p} = l_{p} with equality of (quasi)-norms.

Moreover, for the same range of indices p, q, we shall use the entropy ideals L_{p,q}^{(e)}, formed by all bounded linear operators T between Banach spaces such that the quasi-norm

L_{p,q}^{(e)}(T) := ∥(e_{n}(T))∥_{p,q}

is finite. The classes (L_{p,q}^{(e)}, L_{p,q}^{(e)}) are quasi-normed operator ideals in the sense of Pietsch [21, Definition 6.1.1].

Given any sequence σ = (σ_{k})_{k=1}^{∞}, the diagonal operator D_{σ} is defined by

D_{σ}(x_{k}) := (σ_{k}x_{k}).  (4)

Throughout the paper we shall assume that σ_{1} ≥ σ_{2} ≥ ⋯ ≥ 0. We state now the result of Carl [2] on the entropy numbers of diagonal operators.

Lemma 3. Let 1 ≤ p, q ≤ ∞, 0 < r, s < ∞ and 0 < t ≤ ∞ such that 1/s := 1/r + 1/p − 1/q > 0. Then one has the equivalence

σ = (σ_{k}) ∈ l_{r,t} ⟺ D_{σ} ∈ L_{s,t}^{(e)}(l_{p}, l_{q}).

Remark 4. In the proof the following estimate, for s > 0 with 1/s > 1/p − 1/q > 0,

L_{s,∞}^{(e)} (id : l_{p}^{n} → l_{q}^{n}) ∼ n^{1/s+1/q−1/p}  (5)

was used, which in turn follows from (1) and the elementary properties of entropy numbers.

b. Gaussian measures and entropy. In this subsection all spaces are assumed to be real. The l-norm (introduced by Linde and Pietsch [19] under the name of π_{γ}-norm) of an operator T from a separable Hilbert space H into a Banach space X is defined as

l(T) = sup_{1≤i≤n} (𝔼∥ ∑_{i=1}^{n} T e_{i}g_{i}∥^{2})^{1/2} ,

where (e_{i}) is an arbitrary orthonormal basis in H and (g_{i}) a sequence of i.i.d. standard normal random variables. The l-norm is closely related to Gaussian measures on Banach spaces (see, e.g., the monograph by Ledoux and Talagrand [16]) and is an important tool in Banach space geometry (see the book by Pisier [20]). We only need two basic and well-known facts, the first one can be found, e.g., in [19, Theorem 9].

Lemma 4. There are constants 0 < c < C < ∞ such that the estimate

c · sup_{n∈ℕ}σ_{n}\sqrt{1 + log n} ≤ l(D_{σ}) ≤ C · sup_{n∈ℕ}σ_{n}\sqrt{1 + log n}

holds for every diagonal operator D_{σ} : l_{2} → l_{∞}, with σ_{1} ≥ σ_{2} ≥ ⋯ ≥ 0.

The next result is known as Sudakov’s minoration principle, see [23] for the original formulation, [16, Theorem 3.18] for a variant in the language of Gaussian processes, or [13] for the version given below.

Lemma 5. There is a constant c > 0 such that the inequality

sup_{k∈ℕ}k^{1/2}e_{k}(T^{∗}) ≤ c · l(T)

holds for all operators T from a Hilbert space H into a Banach space X. (Here T^{∗} denotes the dual of T.)

The two preceding lemmas immediately imply

Lemma 6. There is a constant c > 0 such that for the diagonal operator D_{σ} : l_{1} → l_{2}, generated by σ_{k} = (1 + log k)^{−1/2}, the inequality

e_{k}(D_{σ}) ≤ c · k^{−1/2}

holds for all k ∈ ℕ.

Remark 5. By standard considerations one can easily show that this result remains true for complex spaces as well.

c. Complex interpolation and entropy. In this subsection we shall consider only complex Banach spaces since the complex interpolation method does not make sense for real spaces. Our general references to interpolation theory, in particular to the complex method, are the monographs by Bergh and Löfström [1], and Triebel [26]; we shall use the same notation and terminology, and state some known facts for later applications.

First we fix some notation. If 0 < θ < 1 and (A_{0}, A_{1}) is an interpolation couple (i.e., a pair of Banach spaces which both embed linearly and continously into a common Hausdorff vector space), then [A_{0}, A_{1}]_{θ} denotes as usual the complex interpolation space of the couple.

If A is a Banach space and λ > 0 a positive real number, then λA means the same Banach space A, but equipped with the norm

∥x∥_{λA} := λ · ∥x∥_{A} .

Given 1 ≤ p ≤ ∞ and Banach spaces A_{n}, n ∈ ℕ, we denote by l_{p}(A_{n}) the set of all sequences x = (x_{n})_{n}^{∞} such that x_{n} ∈ A_{n} for all n ∈ ℕ and

(∥x_{n}∥_{A_{n}})_{n=1}^{∞} ∈ l_{p} .

The vector-valued spaces l_{p}(A_{n}) are Banach spaces with respect to the norm

∥x∥ := ∥(∥x_{n}∥_{A_{n}})_{n=1}^{∞}∥_{p}.

In the particular case A_{n} = w_{n}ℂ, with w_{n} > 0 for all n ∈ ℕ, we obtain weighted l_{p}-spaces, simply denoted by l_{p}(w), where w = (w_{n})_{n=1}^{∞} is the weight sequence. The following known interpolation formula for weighted l_{p}-spaces will be needed later.

Lemma 7. Let 1 ≤ p_{0}, p_{1} ≤ ∞, but not p_{0} = p_{1} = ∞, let 0 < θ < 1, and let w_{0} and w_{1} be arbitrary weight sequences. Then one has with equality of norms

[l_{p_{0}}(w_{0}), l_{p_{1}}(w_{1})]_{θ} = l_{p}(w_{0}^{1−θ}w_{1}^{θ}), where \frac{1}{p} = \frac{1 − θ}{p_{0}} + \frac{θ}{p_{1}} .  (6)

Proof. From the very definition of the complex method it follows that for every Banach space A and arbitrary real constants λ_{0}, λ_{1} > 0 one has equality of norms

[λ_{0}A, λ_{1}A]_{θ} = λA , where λ = λ_{0}^{1−θ}λ_{1}^{θ}.  (7)

Moreover, given an arbitrary sequence of Banach couples (A_{n}, B_{n}), n ∈ ℕ, the following interpolation formula for vector-valued l_{p}-spaces

[l_{p_{0}}(A_{n}), l_{p_{1}}(B_{n})]_{θ} = l_{p}([A_{n}, B_{n}]_{θ}), where \frac{1}{p} = \frac{1 − θ}{p_{0}} + \frac{θ}{p_{1}} ,  (8)

holds, also with equality of norms, see, e.g., [26, Theorem 1.18.1. and Remarks 1 and 2, p. 122]. From (8) combined with (7), assertion (6) follows.

Finally let us recall the nice behaviour of entropy numbers under interpolation, specified to the complex method, which is sufficient for our present purposes. The general result can be found in [21, Theorems 12.1.11 and 12.1.12]. Here L(X, Y) stands for the class of all bounded linear operators from X into Y.

Lemma 8. Let A and B be Banach spaces, (A_{0}, A_{1}) and (B_{0}, B_{1}) interpolation couples, 0 < θ < 1 and m, n ∈ ℕ. Then the following holds:

(i) If T ∈ L(A_{0} + A_{1}, B), then

e_{m+n−1}(T : [A_{0}, A_{1}]_{θ} → B) ≤ 2e_{m}(T : A_{0} → B)^{1−θ}e_{n}(T : A_{1} → B)^{θ}.

(ii) If T ∈ L(A, B_{0} ∩ B_{1}), then

e_{m+n−1}(T : A → [B_{0}, B_{1}]_{θ}) ≤ 2e_{m}(T : A → B_{0})^{1−θ}e_{n}(T : A → B_{1})^{θ}.

3. Main Results

Now we state our main results on the entropy numbers of diagonal operators. The first two theorems deal with the case, where the diagonal decreases essentially as a power.

Theorem 1. Let 1 ≤ p, q ≤ ∞, α > max(1/q − 1/p, 0) and δ ∈ ℝ. Then one has the implication

σ_{n} ∼ n^{−α}(1 + log n)^{δ} ⟹ e_{n} (D_{σ} : l_{p} → l_{q}) ∼ n^{1/q−1/p−α}(1 + log n)^{δ}.

Theorem 2. Under the assumptions of Theorem 1 one has the equivalence

σ_{n} ⪯ n^{−α}(1 + log n)^{δ} ⟺ e_{n} (D_{σ} : l_{p} → l_{q}) ⪯ n^{1/q−1/p−α}(1 + log n)^{δ}.

It seems appropriate to give here some brief comments on the relation of our results (Theorems 1 and 2) and Carl’s result [2] (stated in Lemma 3). The main difference lies in the way of measuring the rate of decay of a positive decreasing sequence σ = (σ_{n}) of real numbers. While in [2] membership to a Lorentz space l_{r,t} was used, we prefer to work instead with the more direct conditions σ_{n} ⪯ n^{−α}(1 + log n)^{δ}. Both ways provide a two-parameter scale of conditions with the ‘main’ parameter (r, resp. α) and the ‘fine’ parameter (t, resp. δ). Although quite close to each other, the two scales are different, their only ‘intersection point’ being the case (σ_{n}) ∈ l_{r,∞} ⟺ σ_{n} ⪯ n^{−1/r}. In a certain sense our scale is more sensible. For example, the Lorentz scale cannot distuingish between the conditions σ_{n} ⪯ n^{−α}(1 + log n)^{δ}, where α > 0 is fixed and δ > 0 is arbitrary.

Now we turn to the case of logarithmic decay of the diagonal σ, which corresponds to the limiting situation α = 0 in the preceding theorem. Of course, in order to obtain a bounded operator from l_{p} into l_{q}, we have to require p ≤ q. For p = q the entropy behaviour of D_{σ} is known, see Remark 6 below.

Theorem 3. Let 1 ≤ p < q ≤ ∞ and δ > 0, and set δ^{∗} := 1/p − 1/q. Then the condition σ_{n} ∼ (1 + log n)^{−δ} implies

e_{n}(D_{σ} : l_{p} → l_{q}) ∼ \{ \begin{matrix}n^{1/q−1/p}(1 + log n)^{1/p−1/q−δ} & if & δ ≥ δ^{∗} \\ n^{−δ} & if & δ ≤ δ^{∗}. \end{matrix}

Theorem 4. Under the assumptions of Theorem 3 the condition σ_{n} ⪯ (1 + log n)^{−δ} is equivalent to

e_{n}(D_{σ} : l_{p} → l_{q}) ⪯ \{ \begin{matrix}n^{1/q−1/p}(1 + log n)^{1/p−1/q−δ} & if & δ ≥ δ^{∗} \\ n^{−δ} & if & δ ≤ δ^{∗}. \end{matrix}

Remark 6. Note that in the case p = q, δ > 0, σ_{n} ∼ (1 + log n)^{−δ} the result of Gordon, König and Schütt [11], stated here as Lemma 2, implies

e_{n}(D_{σ} : l_{p} → l_{p}) ∼ (1 + log n)^{−δ},

which in contrast to Theorem 2 means that we have only logarithmic decay of the entropy numbers. This shows that the embeddding l_{p} ↪ l_{q} contributes essentially to the power-like decay of the entropy numbers.

Another new feature of the entropy behaviour of diagonal operators with logarithmic diagonals, compared to power-like diagonals, is the existence of the ‘critical’ value δ^{∗}, where the asymptotic entropy behaviour changes drastically. For δ ≥ δ^{∗} the entropy numbers keep almost stable, up to logarithmic factors. A similar phenomenon has occured recently in the study of entropy numbers of convex hulls, see the papers by Carl, Kyrezi and Pajor [3], Gao [10], and Creutzig and Steinwart [6].

4. Proofs

Now we give the proofs of our main results. Proof of Theorem 1. Upper estimate:

For all integers n ≥ 0 consider the dyadic parts D_{n} of the operator D_{σ}, defined by

D_{n}x = (0, …, 0, σ_{2^{n}}x_{2^{n}}, …, σ_{2^{n+1}−1}x_{2^{n+1}−1}, 0, …).

Then, clearly,

D_{σ} = ∑_{n=0}^{∞} D_{n}.  (9)

Choose s > 0 with 1/s > α+1/p−1/q, and observe that the quasi-norm L_{s,∞}^{(e)} is equivalent to an r-norm, for appropriate 0 < r < 1. Now, for any fixed N ∈ ℕ, it follows from (5) and the multiplicativity of entropy numbers that

L_{s,∞}^{(e)} (∑_{n=0}^{N} D_{n})^{r} ≤ c ∑_{n=0}^{N} L_{s,∞}^{(e)}(D_{n})^{r} ≤ c ∑_{n=0}^{N} σ_{2^{n}}^{r}L_{s,∞}^{(e)}(id : l_{p}^{2^{n}} → l_{q}^{2^{n}})^{r}

∼ ∑_{n=0}^{N} (2^{−αn}n^{δ} · 2^{(1/s+1/q−1/p)n})^{r} ∼ (N^{δ}2^{(1/s+1/q−1/p−α)N} )^{r} .

This implies in particular

e_{2^{N}} (∑_{n=0}^{N} D_{n}) ⪯ N^{δ}2^{(1/q−1/p−α)N}.  (10)

Now we estimate the entropy numbers of the tail of the sum (9). To this end, we choose now s > 0 with 1/p − 1/q < 1/s < α + 1/p − 1/q. Again, we use the fact that the quasi-norm L_{s,∞}^{(e)} is equivalent to an r-norm, and so we can estimate it as above (that we have now different r and s does not matter, this influences only the constants):

L_{s,∞}^{(e)} ( ∑_{n=N+1}^{∞} D_{n})^{r} ≤ c ∑_{n=N+1}^{∞} L_{s,∞}^{(e)}(D_{n})^{r}

≤ c ∑_{n=N+1}^{∞} σ_{2^{n}}^{r}L_{s,∞}^{(e)}(id : l_{p}^{2^{n}} → l_{q}^{2^{n}})^{r}

∼ ∑_{n=N+1}^{∞} (2^{−αn}n^{δ} · 2^{(1/s+1/q−1/p)n})^{r} ∼ (N^{δ}2^{(1/s+1/q−1/p−α)N} )^{r} .

It follows that

e_{2^{N}} ( ∑_{n=N+1}^{∞} D_{n}) ⪯ N^{δ}2^{(1/q−1/p−α)N},  (11)

and by the additivity of entropy numbers we conclude from (10) and(11)

e_{2·2^{N}} (D_{σ}) ≤ e_{2^{N}} (∑_{n=0}^{N} D_{n}) + e_{2^{N}} ( ∑_{n=N+1}^{∞} D_{n}) ⪯ N^{δ}2^{(1/q−1/p−α)N},

which proves the upper estimate since N ∈ ℕ was arbitrary.

Lower estimate. For arbitrary n ∈ ℕ, let R_{n} denote the restriction of D_{σ} to the coordinates 1, …, n. Then the multiplicativity of entropy numbers and (1) imply the desired relation

e_{n}(D_{σ}) ≥ e_{n}(R_{n}) ≥ σ_{n} · e_{n}(id : l_{p}^{n} → l_{q}^{n}) ∼ n^{1/q−1/p−α}(1 + log n)^{δ} .

Proof of Theorem 2. The implication ⇒ is covered by Theorem 1. For the converse implication ⇐ we proceed similarly to the lower estimate in the proof of Theorem 1. We now have

n^{1/q−1/p−α}(1 + log n)^{δ} ⪰ e_{n}(D_{σ}) ≥ e_{n}(R_{n})
≥ σ_{n} · e_{n}(id : l_{p}^{n} → l_{q}^{n}) ∼ σ_{n} · n^{1/q−1/p},

whence σ_{n} ⪯ n^{−α}(1 + log n)^{δ}.

Before passing to the proof of Theorem 3 we give an auxiliary result.

Lemma 9. Let w = (w_{n}) be a non-increasing weight sequence, let 1 ≤ p < q ≤ ∞ and set λ := 1/p − 1/q. Then for all n ∈ ℕ

e_{n}(D_{w^{λ}} : l_{p} → l_{q}) ≤ 4e_{n}(D_{w} : l_{1} → l_{∞})^{λ}.

Proof. Let us start with the following simple but useful observation. It is obvious that the Banach spaces l_{p} and l_{p}(w), 1 ≤ p ≤ ∞, w = (w_{n}) an arbitrary weight sequence, are isometrically isomorphic, where the isomorphism is given by the diagonal operator D_{w}, D_{w}(x_{n}) = (w_{n}x_{n}). For the entropy numbers this implies

e_{n}(id : l_{p} → l_{q}(w)) = e_{n}(D_{w} : l_{p} → l_{q}) = e_{n}(id : l_{p}(w^{−1}) → l_{q}),

which means the weights can be shifted from the spaces to the operators. Let us assume for the moment that 1 < p < q < ∞. Step 1. Setting θ := 1/p, we clearly have θ ∈ (0, 1) and \frac{1−θ}{∞} + \frac{θ}{1} = \frac{1}{p} whence

by Lemma 7

[l_{∞}, l_{1}(w^{−1})]_{θ} = l_{p}(w^{−θ}).

Moreover, by shifting the weights Lemma 8 implies

e_{n}(id : l_{p} → l_{∞}(w^{θ})) = e_{n}(id : l_{p}(w^{−θ}) → l_{∞})
≤ 2∥id : l_{∞} → l_{∞}∥^{1−θ}e_{n}(id : l_{1}(w^{−1}) → l_{∞})^{θ}
= 2e_{n}(D_{w} : l_{1} → l_{∞})^{θ}.

Step 2. Next we set η := p/q. Then η ∈ (0, 1) and \frac{1−η}{∞} + \frac{η}{p} = \frac{1}{q} whence Lemma 7 now gives

[l_{∞}(w^{θ}), l_{p}]_{η} = l_{q}(w^{θ(1−η)}) = l_{q}(w^{λ}).

Therefore Lemma 8 and Step 1 yield

e_{n}(D_{w^{λ}} : l_{p} → l_{q}) = e_{n}(id : l_{p} → l_{q}(w^{λ}))
≤ 2e_{n}(id : l_{p} → l_{∞}(w^{θ}))^{1−η}∥id : l_{p} → l_{p}∥^{η}
= 2e_{n}(id : l_{p} → l_{∞}(w^{θ}))^{1−η} ≤ 4e_{n}(D_{w} : l_{1} → l_{∞})^{λ}.

If p = 1 (resp. q = ∞), then the first (resp. second) step can be omitted, and so we have the desired result in all cases.

Proof of Theorem 3. We adopt the notation L_{δ} for the diagonal operator D_{σ} generated by the logarithmic sequence σ_{n} = (1 + log n)^{−δ} with δ > 0, i.e.,

L_{δ}(x_{n}) = ((1 + log n)^{−δ}x_{n}).

Upper estimate: We divide the proof into several steps.

Step 1. First we consider the special case p = 1, q = ∞, δ = 1. By Lemma 6 we have

e_{n}(L_{1/2} : l_{1} → l_{2}) ⪯ n^{−1/2},

and by Tomczak–Jaegermann’s results [24] on the duality of entropy numbers this implies

e_{n}(L_{1/2} : l_{2} → l_{∞}) ⪯ n^{−1/2},

therefore the multiplicativity of entropy numbers yields

e_{2n}(L_{1} : l_{1} → l_{∞}) ≤ e_{n}(L_{1/2} : l_{1} → l_{2}) · e_{n}(L_{1/2} : l_{2} → l_{∞}) ⪯ n^{−1},

which is the desired upper estimate in our special case.

Step 2. Next we consider the case p = 1, q = ∞, δ > 1. Here we write δ = α + 1; clearly, α > 0. By the result of Gordon, König and Schütt [11], see our Remark 6, we have

e_{n}(L_{α} : l_{1} → l_{1}) ∼ (1 + log n)^{−α},

which leads, together with Step 1 and the multiplicativity of the entropy numbers, to the estimate

e_{2n}(L_{δ} : l_{1} → l_{∞}) ≤ e_{n}(L_{α} : l_{1} → l_{1})e_{n}(L_{1} : l_{1} → l_{∞}) ⪯ n^{−1} · (1 + log n)^{−α}.

This is the assertion since −α = 1/p − 1/q − δ in the present case.

Step 3. In the case p = 1, q = ∞, 0 < δ < 1 we use the interpolation formula

[l_{1}, l_{1}(w^{−1})]_{δ} = l_{1}(w^{−δ})

with the special weight sequence w = (w_{n}), w_{n} = 1 + log n. From the result in Step 1 for δ = 1 we conclude analogously to the proof of Lemma 9 that

e_{n}(L_{δ} : l_{1} → l_{∞}) = e_{n}(id : l_{1}(w^{−δ}) → l_{∞})
≤ 2∥id : l_{1} → l_{∞}∥^{1−δ}e_{n}(id : l_{1}((w^{−1}) → l_{∞})^{δ}
= 2e_{n}(L_{1} : l_{1} → l_{∞})^{δ} ⪯ n^{−δ}.

Step 4. Finally, in the general case 1 ≤ p < q ≤ ∞, δ > 0 the assertion follows from Interpolation Lemma 9 and the results in Steps 1–3.

Lower estimate: For arbitrary n ∈ ℕ, let R_{n} denote the restriction of D_{σ} to the coordinates 1, …, n^{2}. Then relation (1) and the multiplicativity of entropy numbers imply

e_{n}(D_{σ}) ≥ e_{n}(R_{n}) ≥ σ_{n^{2}} · e_{n}(id : l_{p}^{n^{2}} → l_{q}^{n^{2}})

∼ (1 + log n)^{−δ} · (\frac{1 + log n}{n})^{1/p−1/q},

and this shows the lower estimate for δ ≥ δ^{∗}. In the case 0 < δ < δ^{∗} we consider the restriction of D_{σ} to the coordinates 1, …, 2^{n}. Then σ_{2^{n}} ∼ n^{−δ} and, similarly to the above we obtain the desired lower estimate

e_{n}(D_{σ}) ≥ σ_{2^{n}} · e_{n}(id : l_{p}^{2^{n}} → l_{q}^{2^{n}}) ∼ n^{−δ}.

Proof of Theorem 4. Theorem 4 can be derived from Theorem 3 in the same way as Theorem 2 from Theorem 1 and therefore the details are omitted.

Acknowledgement

The author greatfully acknowledges the hospitality of Florida Atlantic University, Boca Raton, where the first draft of this paper was written.

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(Received 15.07.2000; revised 22.03.2001)

Author’s address: Mathematisches Institut Universität Leipzig Augustusplatz 10/11, Leipzig Germany E-mail: kuehn@mathematik.uni-leipzig.de