Geometric interpolation of entropy numbers

Geometric interpolation of entropy numbers Abstract We investigate whether the entropy numbers of an operator behave well under the complex interpolation between Hilbert spaces. We study geometric interpolation of entropy numbers of operators. We deliver a contribution to this problem by showing that an interpolation type inequality holds for these numbers. This is in contrast to the situation in Banach spaces, where by examples of Edmunds and Netrusov the entropy numbers do not interpolate well at least in the situation of the real method. As an application, we also present an interpolation estimate of single eigenvalues by single entropy numbers. 1. Introduction The theory of entropy numbers plays a fundamental role in the study of operators and the local theory of Banach spaces. Let T:X→Y be an operator between Banach spaces and let n∈N. The nth entropy number εn(T)=εn(T:X→Y) is defined to be the infimum of all ε>0 such that there exist y1,…,yn∈Y for which T(UX)⊂⋃j=1n{yj+εUY}, where UX denotes the closed unit ball of X. It has turned out that the entropy numbers of an operator are useful in the analysis of the asymptotic behaviour of eigenvalues. The famous inequality due to Carl and Triebel [6] gives an estimate of eigenvalues by single entropy numbers (∏i=1n∣λi(T)∣)1/n≤infk∈Nk1/(2n)εk(T),n∈N, for every T∈K(X) acting on a complex Banach space X, where {λn(T)} stands for its eigenvalue sequence. Zemánek [23] extended this result to the case where T∈L(X). In particular, for k=2n−1, we obtain Carl’s estimate ∣λn(T)∣≤2en(T), en(T)≔ε2n−1(T) denoting the dyadic entropy numbers. We would also draw attention to a long-standing famous problem in the theory of interpolation whether compactness is preserved by interpolation. For the real method, a positive answer has been given by Cwikel in [10] and Cobos et al. in [9], while for the complex method, the problem is still open. Since the entropy numbers and their speed of convergence provide a quantitative way to measure the ‘degree of compactness’ of an operator between Banach spaces, these results motivated the long-open question: how do the entropy numbers of an operator behave under interpolation? A positive answer to this question would lead to numerous valuable applications in the local theory of Banach spaces, including interpolation estimates of eigenvalues by single entropy numbers. Remarkable advances have been made recently in the understanding of the behaviour of entropy numbers. In [12, 13], Edmunds and Netrusov showed that in certain circumstances the entropy numbers of an operator do not behave well under real interpolation. They proved that, given any θ∈(0,1) and q∈[1,∞], there are no positive constants α, C such that for all Banach couples (A0,A1), (B0,B1) and every operator T:(A0,A1)→(B0,B1), the inequality εkα(T:(A0,A1)θ,q→(B0,B1)θ,q)≤Cmax{εk(T:A0→B0),εk(T:A1→B1)} is valid for every k∈N. Nevertheless, it has been shown in many cases (see, e.g. [8, 14, 15, 19, 20]) that the following interpolation type inequality for the measure of non-compactness β(T)≔limn→∞εn(T) holds β(T:F(A0,A1)→F(B0,B1))≤Cφ(β(T:A0→B0),β(T:A1→B1)), where F(Y0,Y1)=[Y0,Y1]θ or F(Y0,Y1)=(Y0,Y1)θ,q (respectively, F(Y0,Y1)=(Y0,Y1)E) with φ(s,t)∼s1−θtθ (respectively, φ(s,t)∼φE(s,t), where φE is a fundamental function of the parameter E generated by the K-method space), s,t≥0. It is vital to bear in mind that the behaviour of entropy numbers under complex interpolation is still unknown, in general. Does this imply that for every θ∈(0,1), there exists a constant C>0 such that for all Banach couples (A0,A1), (B0,B1) and every operator T:(A0,A1)→(B0,B1) the following interpolation inequality holds: εk0k1(T:[A0,A1]θ→[B0,B1]θ)≤Cεk0(T:A0→B0)1−θεk1(T:A1→B1)θ (E) for all k0,k1∈N? It must be mentioned, too, that results of this form are already known in the case where one end point space is fixed (see, e.g. [18, 22]): εk0k1(T:X→B)≤Cεk0(T:A0→B)1−θεk1(T:A1→B)θ,εk0k1(T:A→Y)≤Cεk0(T:A→B0)1−θεk1(T:A→B1)θ, in the case where a Banach space X belongs to the class CK(θ;A⃗) and B=B0=B1 or A=A0=A1 and a Banach space Y belongs to the class CJ(θ;B⃗). However, there is a lack of general results of type of (E) in the available literature, and only a few results of this type have been published before (see [21]). In this paper, we deliver a variant of the above estimate in the case of Hilbert spaces. We actually prove directly that the behaviour of entropy numbers (and inner entropy numbers as well) under geometric interpolation is quite as good as it was conjectured. In fact, we show that for all Hilbert couples (H0,H1), (K0,K1) and every operator T:(H0,H1)→(K0,K1), the estimate εk(T:[H0,H1]θ→[K0,K1]θ)≤72εk(T:H0→K0)1−θεk(T:H1→K1)θ holds for all θ∈(0,1) and k∈N. In case θ=1/2, we may replace the factor 72 by 12. As an example of the application of the main theorem, we present the estimate of single eigenvalues of T acting on [H0,H1]θ by single dyadic entropy numbers of T acting on A0 and A1, namely ∣λn(T:[H0,H1]θ→[H0,H1]θ)∣≤2en(T:H0→H0)1−θen(T:H1→H1)θ which holds for all θ∈(0,1) and n∈N. 2. A refined Riesz theory We shall keep this paper clearer and self-contained by recalling some basic concepts and results from the spectral theory of operators which already appeared in [17, 21]. Given a bounded linear operator T∈L(X) on a complex Banach space X, T is said to be a Fredholm operator provided that its kernel, N(T), is finite dimensional and the range R(T) has a finite codimension. T is a Fredholm operator if and only if its equivalence class is invertible in the Calkin algebra L(X)/K(X), where K(X) denotes the Banach algebra of compact operators on X. We denote by σ(T) the spectrum of T. The essential spectrum σess(T) is the collection of all λ∈C satisfying λIX−T is not Fredholm. The essential spectral radius is given by the formula ress(T)≔sup{∣λ∣:λ∈σess(T)}. We call T∈L(X) with ress(T)=0 a Riesz operator. Examples of Riesz operators are power compact operators. Among the elements of σ(T), the eigenvalues of T are of particular interest. The Fredholm theory provides that the set Λ(T)={λ∈σ(T):∣λ∣>ress(T)} is at most countable and consists of isolated eigenvalues of finite algebraic multiplicity. Following the Riesz theory for compact operators (see [5, 23] for more details), for a given T∈L(X), we can assign an eigenvalue sequence {λn(T)}n=1∞ from the elements of the set Λ(T)∪{ress(T)} as follows: the eigenvalues are arranged in an order of non-increasing absolute values, and each eigenvalue is counted according to its algebraic multiplicity; if T has less than n eigenvalues λ with ∣λ∣>ress(T), we let λn(T)=λn+1(T)=⋯=ress(T); the order could be non-uniquely determined; we choose a fixed order of this form. In this framework, we also have that (see e.g. [5] for more details) λn(Tm)=λn(T)m,m,n∈N (2.1) and λn(RS)=λn(SR),n∈N, (2.2) where R∈L(E,F), S∈L(F,E) are operators acting between complex Banach spaces. 3. Properties of entropy numbers In this section, the reader will be reminded of some important properties of entropy and approximation numbers, and some auxiliary results will be quoted or derived. Let us recall some classical quantities of an operator T∈L(X,Y) acting between Banach spaces (see e.g. [5] for more details). Given n∈N, the nth inner entropy number φn(T)=φn(T:X→Y) is defined to be the supremum of all ρ>0 such that there exist x1,…,xp∈UX, p>n such that ‖T(xi−xj)‖>2ρ,1≤i<j≤p, the nth approximation number an(T)=an(T:X→Y)≔inf{∥T−S∥:S∈L(X,Y),rankS<n}. The next crucial results come from [17, Theorem 3.2 and Proposition 3.6]. Theorem 3.1 Let X be a complex Banach space and T∈L(X). If {λn(T)}is an eigenvalue sequence of T, then limm→∞εkm(Tm)1/m=supn∈Nk−1/(2n)(∏i=1n∣λi(T)∣)1/n,k∈N. Following [17], we define for every operator T on a complex Banach space X the nth spectral entropy number as follows: En(T)≔limm→∞εnm(Tm)1/m for every operator T on a complex Banach space X. Clearly, En(T)≤εn(T) for each n∈N. The following proposition is a direct consequence of Theorem 3.1. One of the appealing aspects of the spectral entropy numbers is that it readily lends itself to explicit computation in the spirit of (2.1) and (2.2): Proposition 3.2 Let X and Y be arbitrary complex Banach spaces. If R∈L(X)and S∈L(X)are commuting operators, then Ekn(RS)≤Ek(R)En(S),k,n∈N.If R∈L(X), then Ekm(Rm)=Ek(R)m,k,m∈N.Moreover, if T∈L(X,Y)and U∈L(Y,X), then Ek(TU)=Ek(UT),k∈N. Let us recall that in the case where T∈L(H) is a normal operator acting on a Hilbert space H, we have the following equality (see [5, Proposition 4.4.1] and also [21, Proposition 3.4]) an(T)=∣λn(T)∣,n∈N and so Ek(T)=supn∈Nk−1/(2n)(∏i=1nai(T))1/n by Theorem 3.1. In our study in this paper, we will use the following theorem. The proof is a combination of [21, Theorem 3.8] and formulae [21, (3.3)]. Theorem 3.3 Let H,Kbe complex Hilbert spaces and T∈L(H,K). Then φn(T:H→K)≤4En(∣T∣:H→H)andεn(T:H→K)≤6En(∣T∣:H→H),n∈N. It is perhaps worth noting that the latter inequality is a restatement of [5, Theorem 3.4.2] (cf. [21, Theorem 3.10]) in terms of the spectral entropy numbers En(T), where the constant appearing at the right-hand side of the inequality is equal to 14. We conclude this section with the following two lemmas. Lemma 3.4 Let E,Fbe arbitrary Banach spaces and T∈L(E,F). Suppose that there exist operators {Pn}n∈Nand {Qn}n∈Nwhich have norm less or equal to 1 and approximate identity on finite subsets of E and F, respectively. Then φk(T:E→F)=limn→∞φk(QnTPn:E→F),k∈N. Proof Note that limsupn→∞φk(QnTPn:E→F)≤φk(T:E→F) holds trivially. Since the operators Pn and Qn approximate identity on finite subsets of E and F, respectively, it follows that ‖(T−QnTPn)x‖F≤‖(I−Qn)Tx‖F+‖QnT(I−Pn)x‖F≤‖(I−Qn)Tx‖F+‖T‖E→F‖(I−Pn)x‖E→E→0 as n→∞, and hence ‖Tx‖F=limn→∞‖QnTPnx‖F for all x∈E. Fix ε>0. By definition, there exists a set of elements x1,…,xn+1∈UE such that 2φk(T:E→F)−ε<‖T(xi−xj)‖F,1≤i<j≤n+1, and an integer N such that ‖T(xi−xj)‖F−ε<‖QnTPn(xi−xj)‖F,n>N,1≤i<j≤n+1. Hence, by definition again, we have min1≤i<j≤n+1‖QnTPn(xi−xj)‖F<2φk(QnTPn)+ε,n>N, and the inequality φk(T:E→F)≤liminfn→∞φk(QnTPn:E→F) follows.□ Until now, we considered entropy numbers εn(T) and φn(T) as a function of any operator T acting between arbitrary Banach spaces. Here and subsequently, we will sometimes drop the assumption that these spaces are complete. Lemma 3.5 Let E and F be arbitrary Banach spaces, and T∈L(E,F). Assume that there exist subspaces E0⊂Eand F0⊂Fwhich are dense in E and F, respectively. If T(E0)⊂F0, then φk(T:E→F)=φk(T:E0→F0),εk(T:E→F)=εk(T:E0→F0),k∈N. Proof Fix k∈N. We first show that φk(T:E→F)=φk(T:E0→F0). Let 0<δ<ρ<φk(T:E→F). By definition, there exists a set of elements x1,…,xn+1∈UE such that 2ρ<‖T(xi−xj)‖F,1≤i<j≤n+1. Choose xi0∈UE0 which satisfy ‖Txi0−Txi‖F<δ, 1≤i≤k. By the above, 2ρ<‖T(xi0−xj0)‖F+2δ,1≤i<j≤n+1, and consequently, φk(T:E→F)≤φk(T:E0→F0). It follows immediately that φk(T:E0→F0)≤φk(T:E→F). The equality εk(T:E→F)=εk(T:E0→F0) was shown in the proof of [21, Lemma 3.11].□ 4. Interpolation of entropy numbers In this section, we look at some specific techniques from interpolation theory which can be briefly described as ‘geometric interpolation’ methods. We also develop tools which will be essential in geometric interpolation of the entropy numbers of operators. We start with some elementary definitions from the interpolation theory of operators. We will generally use the same notation as in [3, 4, 16, 21]. The Banach space X will be called an intermediate space between A0 and A1(or with respect to a Banach couple A⃗≔(A0,A1)) provided A0∩A1⊂X⊂A0+A1. A Banach couple (A0,A1) is called regular if Aj◦=Aj, where Aj◦ denote the closure of Δ(A⃗)≔A0∩A1 in Aj for j=0,1. If A⃗=(A0,A1) and B⃗=(B0,B1) are Banach couples and T:A0+A1→B0+B1 is a linear map such that T∣Aj∈L(Aj,Bj) for j=0,1, then we write T:A⃗→B⃗. The space L(A⃗,B⃗) of all operators T:A⃗→B⃗ is a Banach space equipped with the norm ∥T∥≔maxj=0,1∥T∣Aj∥L(Aj,Bj). Banach spaces X and Y are said to be interpolation spaces with respect to A⃗ and B⃗ if X and Y are intermediate with respect to A⃗ and B⃗, and if T maps X into Y for every T∈L(A⃗,B⃗). If in addition there exists C>0 and θ∈(0,1) such that ‖T:X→Y‖≤C‖T:A0→B0‖1−θ‖T:A1→B1‖θ for every T∈L(A⃗,B⃗), then X and Y are said to be of exponent θ (and exact of exponent θ if C=1). It is well known that the complex interpolation space [A⃗]θ is exact of exponent θ. We now turn to geometric interpolation between Hilbert spaces. The best general reference here is McCarthy [7] where more details are given (see also Donoghue [11], Ameur [2] or [21] modulo some evident misprints on page 356, where it should be RanA1/2¯ and RanAθ/2¯, respectively). Let H⃗=(H0,H1) be a regular couple of Hilbert spaces. As is well known, by the Riesz representation theorem (see e.g. [11, p. 253] and [1, p. 261] for more details), there exists a unique, positive self-adjoint operator A in H0 such that ⟨ξ,η⟩H1=⟨A1/2ξ,A1/2η⟩H0,ξ,η∈Δ(H⃗), where DomA1/2=Δ(H⃗) and RanA1/2¯=H0. The operator A is bounded if and only if H0 is contained in H1. Let θ∈(0,1). We define a new inner product on Δ(H⃗) by ⟨ξ,η⟩θ=⟨Aθ/2ξ,Aθ/2η⟩H0,θ∈(0,1). Δ(H⃗) is contained in DomAθ/2 and RanAθ/2¯=H0. The closure of Δ(H⃗), with respect to the norm given by the inner product, we will call Hθ. The space Hθ is a geometric interpolation space of exponent θ and coincides with [H⃗]θ. The proof of the following lemma is partially based on arguments introduced in [21, Lemma 4.1], but for the sake of completeness, all essential details are provided. Lemma 4.1 Let H⃗and K⃗be regular couples of Hilbert spaces. Assume that A and B are positive operators on H0and K0that give the H1and K1inner product, respectively. If T∈L(H⃗,K⃗), then φk(T:Hθ→Kθ)=φk(Bθ/2TA−θ/2:H0→K0)andεk(T:Hθ→Kθ)=εk(Bθ/2TA−θ/2:H0→K0),θ∈[0,1],k∈N. Proof Fix θ∈[0,1]. We denote by HθΔ≔(Δ(H⃗),∥·∥Hθ), H0Δ≔(RanAθ/2,∥·∥H0) and KθΔ≔(Δ(K⃗),∥·∥Kθ), K0Δ≔(RanBθ/2,∥·∥K0). It was concluded in the proof of [21, Lemma 4.1] that the closure of S≔Bθ/2TA−θ/2:H0Δ→K0Δ is bounded as a map between H0 and K0, justifying the later use of Bθ/2TA−θ/2 for the extension of S to H0. Let k∈N. We proceed to show that φk(T:HθΔ→KθΔ)=φk(S:H0Δ→K0Δ). Let 0<ρ<φk(S:H0Δ→K0Δ). By definition, there exists a set of elements x1,…,xn+1∈UH0Δ such that 2ρ<‖S(xi−xj)‖K0Δ,1≤i<j≤n+1. As in the proof of [21, Lemma 4.1], we have ‖T(x˜i−x˜j)‖KθΔ=‖S(xi−xj)‖K0Δ,1≤i<j≤n+1, where x˜1,…,x˜n+1∈UHθΔ. Therefore, φk(S:H0Δ→K0Δ)≤φk(T:HθΔ→KθΔ). In the same manner, we can check the reverse inequality. That εk(T:HθΔ→KθΔ)=εk(S:H0Δ→K0Δ) was already verified in the proof of [21, Lemma 4.1]. An application of Lemma 3.5 completes the proof.□ We can now state our main result. Theorem 4.2 Assume that H⃗=(H0,H1)and K⃗=(K0,K1)are arbitrary couples of Hilbert spaces, T∈L(H⃗,K⃗)and k∈N. Then φk(T:[H⃗]θ→[K⃗]θ)≤64φk(T:H0→K0)1−θφk(T:H1→K1)θandεk(T:[H⃗]θ→[K⃗]θ)≤72εk(T:H0→K0)1−θεk(T:H1→K1)θfor every θ∈(0,1). In the special case θ=1/2, the constants can be improved, namely φk(T:[H⃗]1/2→[K⃗]1/2)≤8φk(T:H0→K0)1/2φk(T:H1→K1)1/2andεk(T:[H⃗]1/2→[K⃗]1/2)≤12εk(T:H0→K0)1/2εk(T:H1→K1)1/2. In the case of operators acting between Hilbert spaces, as it will be in most of this paper, we are interested in improving the estimates obtained (possibly with optimal constants). It seems likely that the arguments employed in the proof of Theorem 4.2 would be much more involved here. Therefore, although these proofs (Theorem 4.2 and [21, Theorem 4.2] modulo some obvious misprints) run along similar lines, there are subtle adjustments necessary to fit the argument to each new situation. Proof Here, therefore, it is assumed for simplicity’s sake that H⃗ and K⃗ are regular. Let A (respectively, B) be the positive operator on H0 (respectively, K0) that gives the H1 (respectively, K1) inner product. Fix θ∈[0,1]. That Bθ/2TA−θ/2∈L(H0,K0) follows from Lemma 4.1. To ease notation, set Rθ=Bθ/2TA−θ/2. Fix k∈N. By Lemma 4.1 again, the kth (inner) entropy number of T from Hθ to Kθ equals the kth (inner) entropy number of Rθ from H0 to K0, namely φk(T:Hθ→Kθ)=φk(Rθ:H0→K0)andεk(T:Hθ→Kθ)=εk(Rθ:H0→K0). (4.1) We first prove a reduced form of the theorem for the family of operators Rθ. Suppose that A−1 and B are bounded. From [21, (3.3)] and Proposition 3.2, we obtain Ek(∣R1/2∣:H0→H0)=Ek2(R1/2*R1/2:H0→H0)1/2≤Ek2(A1/4R1/2*R1/2A−1/4:H0→H0)1/2≤εk(T*:K0→H0)1/2εk(B1/2TA−1/2:H0→K0)1/2. Theorem 3.3 now yields the assertion for θ=1/2 with a constant C=4 or D=6, respectively, φk(R1/2:H0→K0)≤2Cφk(R0:H0→K0)1/2φk(R1:H0→K0)1/2andεk(R1/2:H0→K0)≤Dεk(R0:H0→K0)1/2εk(R1:H0→K0)1/2. (4.2) Now interpolating between R0 and R1/2 or R1/2 and R1 gives the result for θ=1/4 or θ=3/4, respectively. The constant here is equal to (2C)3/2 (respectively, D3/2). Following the same lines, we find that the theorem holds for any dyadic rational in [0,1] with a common constant 4C2 (respectively, D2). Indeed, one may check φk(Rθ:H0→K0)≤4C2φk(R0:H0→K0)1−θφk(R1:H0→K0)θandεk(Rθ:H0→K0)≤D2εk(R0:H0→K0)1−θεk(R1:H0→K0)θ (4.3) for any dyadic rational θ=m/2n∈(0,1) by induction on n. That (4.3) is valid for θ=1/2 is already proved in (4.2). For the inductive step, suppose that (4.3) holds for θ=m/2n, 0<m<2n with a constant equal to D2. It suffices to consider θ=m/2n+1 where 0<m<2n+1 is odd. Now interpolating between R0 and R2θ or R2θ−1 and R1 gives (4.3) for θ<1/2 or θ>1/2, respectively. That ‖Rα−Rβ‖H0→K0→0 as α→β was validated in the proof of [21, Lemma 3.11]. Therefore, our claim is valid for any real θ∈[0,1]. This is because ∣φk(Rα:H0→K0)−φk(Rβ:H0→K0)∣≤‖Rα−Rβ‖H0→K0and∣εk(Rα:H0→K0)−εk(Rβ:H0→K0)∣≤‖Rα−Rβ‖H0→K0 by [21, Proposition 3.5]. We have been working under the assumption that both A−1 and B are bounded. Suppose now that this is no longer so. Take a look at the operators Pn≔∫n−1ndEA−1(λ) and Qn≔∫n−1ndEB(λ), where EA−1 and EB are the corresponding spectral projections. The operators Pn and Qn have bounded extensions on H⃗ and K⃗, which are norm 1 projections on Hθ and Kθ, respectively. Thus QnTPn∈L(H⃗,K⃗). Using equalities (4.1) and Bnθ/2TAn−θ/2=Bθ/2QnTPnA−θ/2, where An−1≔∫n−1nλdEA−1(λ) and Bn≔∫n−1nλdEB(λ) are bounded, and following steps analogous to those above (with A−1, B and Rθ replaced by An−1, Bn, and Bnθ/2TAn−θ/2, respectively), we obtain φk(QnTPn:Hθ→Kθ)≤4C2φk(T:H0→K0)1−θφk(T:H1→K1)θandεk(QnTPn:Hθ→Kθ)≤D2εk(T:H0→K0)1−θεk(T:H1→K1)θ for every θ∈[0,1]. Since Pn (respectively, Qn) are approximate identities on finite subsets of Hθ (respectively, Kθ), it follows that φk(T:Hθ→Kθ)≤4C2φk(T:H0→K0)1−θφk(T:H1→K1)θandεk(T:Hθ→Kθ)≤2liminfn→∞εk(QnTPn:Hθ→Kθ)≤2D2εk(T:H0→K0)1−θεk(T:H1→K1)θ, by Lemma 3.4. The same reasoning applies to (4.2). Since entropy numbers are injective in the sense of [21, Proposition 3.7], the case where the couples H⃗ and K⃗ are not necessarily regular can be handled in much the same way as in the proof of [21, Theorem 4.2].□ Perhaps it is appropriate at this point to note that in order to inductively prove the inequality (4.3) for dyadic fractions, we have to interpolate between one of the spaces H0 or H1, just to keep the constants uniformly bounded (cf. [21, Theorem 4.2]). As an example of the application of Theorem 4.2, we derive the following interpolation estimate in the spirit of Carl’s result. Corollary 4.3 Let H⃗=(H0,H1)be a couple of Hilbert spaces. Then for all T∈L(H⃗), θ∈(0,1)and n∈N, we have En(T:[H⃗]θ→[H⃗]θ)≤En(T:H0→H0)1−θEn(T:H1→H1)θ.In particular, ∣λn(T:[H⃗]θ→[H⃗]θ)∣≤2en(T:H0→H0)1−θen(T:H1→H1)θ. Before going to the proof, it is worth noting that the second statement follows directly from Theorem 4.2 and the Carl–Triebel inequality (see [23, Proposition 3.5] or [5, Theorem 4.2.1]). The proof we give is based instead on the spectral properties of entropy numbers, namely supn∈Nk−1/(2n)(∏i=1n∣λi(T)∣)1/n=limm→∞εkm(Tm)1/m. In a nutshell, various variants of (E) may lead to interpolation results on spectral entropy numbers of this kind, and are therefore of considerable interest. Proof Fix θ∈(0,1). Let k,n,m∈N. Theorem 4.2 now yields εk(Tm:[H⃗]θ→[H⃗]θ)≤Cεk(Tm:H0→H0)1−θεk(Tm:H1→H1)θ. We conclude from Theorem 3.1 that Ek(T:[H⃗]θ→[H⃗]θ)≤Ek(T:H0→H0)1−θEk(T:H1→H1)θ, hence that k−1/(2n)(∏i=1n∣λi(T:[H⃗]θ→[H⃗]θ)∣)1/n≤εk(T:H0→H0)1−θεk(T:H1→H1)θ, and the last assertion follows for k=2n−1.□ Let us remark that in the case of normal operators acting on Hilbert spaces, the analogous interpolation result on entropy numbers follows directly from [21, Theorems 3.8 and 4.3]: Theorem 4.4 Assume that H⃗=(H0,H1)is a regular couple of Hilbert spaces. Let A be a positive operator on H0that gives the H1inner product and let T∈L(H⃗). If the operator T on H0is normal and commutes with A, then 1/8φk(T:H0→H0)≤φk(T:Hθ→Hθ)≤8φk(T:H0→H0)and1/6εk(T:H0→H0)≤εk(T:Hθ→Hθ)≤6εk(T:H0→H0)for every θ∈(0,1]and each k∈N. 5. Conclusion The results of the paper are related to a long standing problem on interpolation of entropy numbers. Edmunds and Netrusov [12] proved that in certain circumstances the entropy numbers of an operator do not behave well under (real) interpolation. Indeed, given any θ∈(0,1) and q∈[1,∞], there are no positive constants α, C such that for all Banach couples (A0,A1), (B0,B1) and every operator T:(A0,A1)→(B0,B1), the inequality εkα(T:A⃗θ,q→B⃗θ,q)≤Cmax{εk(T:A0→B0),εk(T:A1→B1)} holds for every k∈N. However, the (complex) interpolation estimate εk(T:[H⃗]θ→[K⃗]θ)≤72εk(T:H0→K0)1−θεk(T:H1→K1)θ proved directly in this work, is the first one of the type related to the mentioned problem, which holds for all operators between all compatible couples of Hilbert spaces. Funding The author was supported by the Foundation for Polish Science (FNP). References 1 Y. Ameur , A new proof of Donoghue’s interpolation theorem , J. Funct. Spaces Appl 2 ( 2004 ), 253 – 265 . Google Scholar CrossRef Search ADS 2 Y. Ameur , Interpolation of Hilbert spaces, U.U.D.M. Thesis 20 ( 2001 ). 3 J. Bergh and J. Löfström , Interpolation Spaces. An Introduction , Springer-Verlag , Berlin , 1976 , Grundlehren der Mathematischen Wissenschaften, No. 223. Google Scholar CrossRef Search ADS 4 Yu. A. Brudny and N. Ya. Krugljak , Interpolation Functors and Interpolation Spaces. Vol. I, North-Holland Mathematical Library Vol. 47 , North-Holland Publishing Co , Amsterdam , 1991 , Translated from the Russian by Natalie Wadhwa, with a preface by Jaak Peetre. 5 B. Carl and I. Stephani , Entropy, Compactness and the Approximation of Operators, Cambridge Tracts in Mathematics Vol. 98 , Cambridge University Press , Cambridge , 1990 . Google Scholar CrossRef Search ADS 6 B. Carl and H. Triebel , Inequalities between eigenvalues, entropy numbers, and related quantities of compact operators in Banach spaces , Math. Ann. 251 ( 1980 ), 129 – 133 . Google Scholar CrossRef Search ADS 7 J. E. McCarthy , Geometric interpolation between Hilbert spaces , Ark. Mat. 30 ( 1992 ), 321 – 330 . Google Scholar CrossRef Search ADS 8 F. Cobos , P. Fernández-Martnez and A. Martnez , Interpolation of the measure of non-compactness by the real method , Studia Math. 135 ( 1999 ), 25 – 38 . 9 F. Cobos , T. Kühn and T. Schonbek , One-sided compactness results for Aronszajn–Gagliardo functors , J. Funct. Anal. 106 ( 1992 ), 274 – 313 . Google Scholar CrossRef Search ADS 10 M. Cwikel , Real and complex interpolation and extrapolation of compact operators , Duke Math. J. 65 ( 1992 ), 333 – 343 . Google Scholar CrossRef Search ADS 11 W. F. Donoghue Jr. , The interpolation of quadratic norms , Acta Math. 118 ( 1967 ), 251 – 270 . Google Scholar CrossRef Search ADS 12 D. E. Edmunds and Yu. Netrusov , Entropy numbers and interpolation , Math. Ann. 351 ( 2011 ), 963 – 977 . Google Scholar CrossRef Search ADS 13 D. E. Edmunds and Yu. Netrusov , Entropy numbers of operators acting between vector-valued sequence spaces , Math. Nachr. 286 ( 2013 ), 614 – 630 . Google Scholar CrossRef Search ADS 14 D. E. Edmunds and M. F. Teixeira , Interpolation theory and measures of noncompactness , Math. Nachr. 104 ( 1981 ), 129 – 135 . Google Scholar CrossRef Search ADS 15 P. Fernández-Martnez , Interpolation of the measure of non-compactness between quasi-Banach spaces , Rev. Mat. Complut. 19 ( 2006 ), 477 – 498 . 16 S. G. Kren , Yu. I. Petunin and E. M. Semenov , Interpolation of Linear Operators, Translations of Mathematical Monographs Vol. 54 , American Mathematical Society , Providence, R.I. , 1982 . Translated from the Russian by J. Szücs. 17 M. Mastyło and R. Szwedek , Eigenvalues and entropy moduli of operators in interpolation spaces , J. Geom. Anal. 27 ( 2017 ), 1131 – 1177 . Google Scholar CrossRef Search ADS 18 A. Pietsch , Operator Ideals, Mathematische Monographien [Mathematical Monographs] vol. 16 , VEB Deutscher Verlag der Wissenschaften , Berlin , 1978 . 19 R. Szwedek , Measure of non-compactness of operators interpolated by the real method , Studia Math. 175 ( 2006 ), 157 – 174 . Google Scholar CrossRef Search ADS 20 R. Szwedek , On interpolation of the measure of non-compactness by the complex method , Q. J. Math. 66 ( 2015 ), 323 – 332 . Google Scholar CrossRef Search ADS 21 R. Szwedek , Interpolation of approximation numbers between Hilbert spaces , Ann. Acad. Sci. Fenn. Math. 40 ( 2015 ), 343 – 360 . Google Scholar CrossRef Search ADS 22 H. Triebel , Interpolation Theory, Function Spaces, Differential Operators , VEB Deutscher Verlag der Wissenschaften , Berlin , 1978 . 23 J. Zemánek , The Essential Spectral Radius and the Riesz Part of the Spectrum, Functions, series, operators, Vols. I and II (Budapest, 1980), Colloq. Math. Soc. János Bolyai Vol. 35 , North-Holland , Amsterdam , 1983 , 1275 – 1289 . © 2017. Published by Oxford University Press. All rights reserved. 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Geometric interpolation of entropy numbers

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Abstract We investigate whether the entropy numbers of an operator behave well under the complex interpolation between Hilbert spaces. We study geometric interpolation of entropy numbers of operators. We deliver a contribution to this problem by showing that an interpolation type inequality holds for these numbers. This is in contrast to the situation in Banach spaces, where by examples of Edmunds and Netrusov the entropy numbers do not interpolate well at least in the situation of the real method. As an application, we also present an interpolation estimate of single eigenvalues by single entropy numbers. 1. Introduction The theory of entropy numbers plays a fundamental role in the study of operators and the local theory of Banach spaces. Let T:X→Y be an operator between Banach spaces and let n∈N. The nth entropy number εn(T)=εn(T:X→Y) is defined to be the infimum of all ε>0 such that there exist y1,…,yn∈Y for which T(UX)⊂⋃j=1n{yj+εUY}, where UX denotes the closed unit ball of X. It has turned out that the entropy numbers of an operator are useful in the analysis of the asymptotic behaviour of eigenvalues. The famous inequality due to Carl and Triebel [6] gives an estimate of eigenvalues by single entropy numbers (∏i=1n∣λi(T)∣)1/n≤infk∈Nk1/(2n)εk(T),n∈N, for every T∈K(X) acting on a complex Banach space X, where {λn(T)} stands for its eigenvalue sequence. Zemánek [23] extended this result to the case where T∈L(X). In particular, for k=2n−1, we obtain Carl’s estimate ∣λn(T)∣≤2en(T), en(T)≔ε2n−1(T) denoting the dyadic entropy numbers. We would also draw attention to a long-standing famous problem in the theory of interpolation whether compactness is preserved by interpolation. For the real method, a positive answer has been given by Cwikel in [10] and Cobos et al. in [9], while for the complex method, the problem is still open. Since the entropy numbers and their speed of convergence provide a quantitative way to measure the ‘degree of compactness’ of an operator between Banach spaces, these results motivated the long-open question: how do the entropy numbers of an operator behave under interpolation? A positive answer to this question would lead to numerous valuable applications in the local theory of Banach spaces, including interpolation estimates of eigenvalues by single entropy numbers. Remarkable advances have been made recently in the understanding of the behaviour of entropy numbers. In [12, 13], Edmunds and Netrusov showed that in certain circumstances the entropy numbers of an operator do not behave well under real interpolation. They proved that, given any θ∈(0,1) and q∈[1,∞], there are no positive constants α, C such that for all Banach couples (A0,A1), (B0,B1) and every operator T:(A0,A1)→(B0,B1), the inequality εkα(T:(A0,A1)θ,q→(B0,B1)θ,q)≤Cmax{εk(T:A0→B0),εk(T:A1→B1)} is valid for every k∈N. Nevertheless, it has been shown in many cases (see, e.g. [8, 14, 15, 19, 20]) that the following interpolation type inequality for the measure of non-compactness β(T)≔limn→∞εn(T) holds β(T:F(A0,A1)→F(B0,B1))≤Cφ(β(T:A0→B0),β(T:A1→B1)), where F(Y0,Y1)=[Y0,Y1]θ or F(Y0,Y1)=(Y0,Y1)θ,q (respectively, F(Y0,Y1)=(Y0,Y1)E) with φ(s,t)∼s1−θtθ (respectively, φ(s,t)∼φE(s,t), where φE is a fundamental function of the parameter E generated by the K-method space), s,t≥0. It is vital to bear in mind that the behaviour of entropy numbers under complex interpolation is still unknown, in general. Does this imply that for every θ∈(0,1), there exists a constant C>0 such that for all Banach couples (A0,A1), (B0,B1) and every operator T:(A0,A1)→(B0,B1) the following interpolation inequality holds: εk0k1(T:[A0,A1]θ→[B0,B1]θ)≤Cεk0(T:A0→B0)1−θεk1(T:A1→B1)θ (E) for all k0,k1∈N? It must be mentioned, too, that results of this form are already known in the case where one end point space is fixed (see, e.g. [18, 22]): εk0k1(T:X→B)≤Cεk0(T:A0→B)1−θεk1(T:A1→B)θ,εk0k1(T:A→Y)≤Cεk0(T:A→B0)1−θεk1(T:A→B1)θ, in the case where a Banach space X belongs to the class CK(θ;A⃗) and B=B0=B1 or A=A0=A1 and a Banach space Y belongs to the class CJ(θ;B⃗). However, there is a lack of general results of type of (E) in the available literature, and only a few results of this type have been published before (see [21]). In this paper, we deliver a variant of the above estimate in the case of Hilbert spaces. We actually prove directly that the behaviour of entropy numbers (and inner entropy numbers as well) under geometric interpolation is quite as good as it was conjectured. In fact, we show that for all Hilbert couples (H0,H1), (K0,K1) and every operator T:(H0,H1)→(K0,K1), the estimate εk(T:[H0,H1]θ→[K0,K1]θ)≤72εk(T:H0→K0)1−θεk(T:H1→K1)θ holds for all θ∈(0,1) and k∈N. In case θ=1/2, we may replace the factor 72 by 12. As an example of the application of the main theorem, we present the estimate of single eigenvalues of T acting on [H0,H1]θ by single dyadic entropy numbers of T acting on A0 and A1, namely ∣λn(T:[H0,H1]θ→[H0,H1]θ)∣≤2en(T:H0→H0)1−θen(T:H1→H1)θ which holds for all θ∈(0,1) and n∈N. 2. A refined Riesz theory We shall keep this paper clearer and self-contained by recalling some basic concepts and results from the spectral theory of operators which already appeared in [17, 21]. Given a bounded linear operator T∈L(X) on a complex Banach space X, T is said to be a Fredholm operator provided that its kernel, N(T), is finite dimensional and the range R(T) has a finite codimension. T is a Fredholm operator if and only if its equivalence class is invertible in the Calkin algebra L(X)/K(X), where K(X) denotes the Banach algebra of compact operators on X. We denote by σ(T) the spectrum of T. The essential spectrum σess(T) is the collection of all λ∈C satisfying λIX−T is not Fredholm. The essential spectral radius is given by the formula ress(T)≔sup{∣λ∣:λ∈σess(T)}. We call T∈L(X) with ress(T)=0 a Riesz operator. Examples of Riesz operators are power compact operators. Among the elements of σ(T), the eigenvalues of T are of particular interest. The Fredholm theory provides that the set Λ(T)={λ∈σ(T):∣λ∣>ress(T)} is at most countable and consists of isolated eigenvalues of finite algebraic multiplicity. Following the Riesz theory for compact operators (see [5, 23] for more details), for a given T∈L(X), we can assign an eigenvalue sequence {λn(T)}n=1∞ from the elements of the set Λ(T)∪{ress(T)} as follows: the eigenvalues are arranged in an order of non-increasing absolute values, and each eigenvalue is counted according to its algebraic multiplicity; if T has less than n eigenvalues λ with ∣λ∣>ress(T), we let λn(T)=λn+1(T)=⋯=ress(T); the order could be non-uniquely determined; we choose a fixed order of this form. In this framework, we also have that (see e.g. [5] for more details) λn(Tm)=λn(T)m,m,n∈N (2.1) and λn(RS)=λn(SR),n∈N, (2.2) where R∈L(E,F), S∈L(F,E) are operators acting between complex Banach spaces. 3. Properties of entropy numbers In this section, the reader will be reminded of some important properties of entropy and approximation numbers, and some auxiliary results will be quoted or derived. Let us recall some classical quantities of an operator T∈L(X,Y) acting between Banach spaces (see e.g. [5] for more details). Given n∈N, the nth inner entropy number φn(T)=φn(T:X→Y) is defined to be the supremum of all ρ>0 such that there exist x1,…,xp∈UX, p>n such that ‖T(xi−xj)‖>2ρ,1≤i<j≤p, the nth approximation number an(T)=an(T:X→Y)≔inf{∥T−S∥:S∈L(X,Y),rankS<n}. The next crucial results come from [17, Theorem 3.2 and Proposition 3.6]. Theorem 3.1 Let X be a complex Banach space and T∈L(X). If {λn(T)}is an eigenvalue sequence of T, then limm→∞εkm(Tm)1/m=supn∈Nk−1/(2n)(∏i=1n∣λi(T)∣)1/n,k∈N. Following [17], we define for every operator T on a complex Banach space X the nth spectral entropy number as follows: En(T)≔limm→∞εnm(Tm)1/m for every operator T on a complex Banach space X. Clearly, En(T)≤εn(T) for each n∈N. The following proposition is a direct consequence of Theorem 3.1. One of the appealing aspects of the spectral entropy numbers is that it readily lends itself to explicit computation in the spirit of (2.1) and (2.2): Proposition 3.2 Let X and Y be arbitrary complex Banach spaces. If R∈L(X)and S∈L(X)are commuting operators, then Ekn(RS)≤Ek(R)En(S),k,n∈N.If R∈L(X), then Ekm(Rm)=Ek(R)m,k,m∈N.Moreover, if T∈L(X,Y)and U∈L(Y,X), then Ek(TU)=Ek(UT),k∈N. Let us recall that in the case where T∈L(H) is a normal operator acting on a Hilbert space H, we have the following equality (see [5, Proposition 4.4.1] and also [21, Proposition 3.4]) an(T)=∣λn(T)∣,n∈N and so Ek(T)=supn∈Nk−1/(2n)(∏i=1nai(T))1/n by Theorem 3.1. In our study in this paper, we will use the following theorem. The proof is a combination of [21, Theorem 3.8] and formulae [21, (3.3)]. Theorem 3.3 Let H,Kbe complex Hilbert spaces and T∈L(H,K). Then φn(T:H→K)≤4En(∣T∣:H→H)andεn(T:H→K)≤6En(∣T∣:H→H),n∈N. It is perhaps worth noting that the latter inequality is a restatement of [5, Theorem 3.4.2] (cf. [21, Theorem 3.10]) in terms of the spectral entropy numbers En(T), where the constant appearing at the right-hand side of the inequality is equal to 14. We conclude this section with the following two lemmas. Lemma 3.4 Let E,Fbe arbitrary Banach spaces and T∈L(E,F). Suppose that there exist operators {Pn}n∈Nand {Qn}n∈Nwhich have norm less or equal to 1 and approximate identity on finite subsets of E and F, respectively. Then φk(T:E→F)=limn→∞φk(QnTPn:E→F),k∈N. Proof Note that limsupn→∞φk(QnTPn:E→F)≤φk(T:E→F) holds trivially. Since the operators Pn and Qn approximate identity on finite subsets of E and F, respectively, it follows that ‖(T−QnTPn)x‖F≤‖(I−Qn)Tx‖F+‖QnT(I−Pn)x‖F≤‖(I−Qn)Tx‖F+‖T‖E→F‖(I−Pn)x‖E→E→0 as n→∞, and hence ‖Tx‖F=limn→∞‖QnTPnx‖F for all x∈E. Fix ε>0. By definition, there exists a set of elements x1,…,xn+1∈UE such that 2φk(T:E→F)−ε<‖T(xi−xj)‖F,1≤i<j≤n+1, and an integer N such that ‖T(xi−xj)‖F−ε<‖QnTPn(xi−xj)‖F,n>N,1≤i<j≤n+1. Hence, by definition again, we have min1≤i<j≤n+1‖QnTPn(xi−xj)‖F<2φk(QnTPn)+ε,n>N, and the inequality φk(T:E→F)≤liminfn→∞φk(QnTPn:E→F) follows.□ Until now, we considered entropy numbers εn(T) and φn(T) as a function of any operator T acting between arbitrary Banach spaces. Here and subsequently, we will sometimes drop the assumption that these spaces are complete. Lemma 3.5 Let E and F be arbitrary Banach spaces, and T∈L(E,F). Assume that there exist subspaces E0⊂Eand F0⊂Fwhich are dense in E and F, respectively. If T(E0)⊂F0, then φk(T:E→F)=φk(T:E0→F0),εk(T:E→F)=εk(T:E0→F0),k∈N. Proof Fix k∈N. We first show that φk(T:E→F)=φk(T:E0→F0). Let 0<δ<ρ<φk(T:E→F). By definition, there exists a set of elements x1,…,xn+1∈UE such that 2ρ<‖T(xi−xj)‖F,1≤i<j≤n+1. Choose xi0∈UE0 which satisfy ‖Txi0−Txi‖F<δ, 1≤i≤k. By the above, 2ρ<‖T(xi0−xj0)‖F+2δ,1≤i<j≤n+1, and consequently, φk(T:E→F)≤φk(T:E0→F0). It follows immediately that φk(T:E0→F0)≤φk(T:E→F). The equality εk(T:E→F)=εk(T:E0→F0) was shown in the proof of [21, Lemma 3.11].□ 4. Interpolation of entropy numbers In this section, we look at some specific techniques from interpolation theory which can be briefly described as ‘geometric interpolation’ methods. We also develop tools which will be essential in geometric interpolation of the entropy numbers of operators. We start with some elementary definitions from the interpolation theory of operators. We will generally use the same notation as in [3, 4, 16, 21]. The Banach space X will be called an intermediate space between A0 and A1(or with respect to a Banach couple A⃗≔(A0,A1)) provided A0∩A1⊂X⊂A0+A1. A Banach couple (A0,A1) is called regular if Aj◦=Aj, where Aj◦ denote the closure of Δ(A⃗)≔A0∩A1 in Aj for j=0,1. If A⃗=(A0,A1) and B⃗=(B0,B1) are Banach couples and T:A0+A1→B0+B1 is a linear map such that T∣Aj∈L(Aj,Bj) for j=0,1, then we write T:A⃗→B⃗. The space L(A⃗,B⃗) of all operators T:A⃗→B⃗ is a Banach space equipped with the norm ∥T∥≔maxj=0,1∥T∣Aj∥L(Aj,Bj). Banach spaces X and Y are said to be interpolation spaces with respect to A⃗ and B⃗ if X and Y are intermediate with respect to A⃗ and B⃗, and if T maps X into Y for every T∈L(A⃗,B⃗). If in addition there exists C>0 and θ∈(0,1) such that ‖T:X→Y‖≤C‖T:A0→B0‖1−θ‖T:A1→B1‖θ for every T∈L(A⃗,B⃗), then X and Y are said to be of exponent θ (and exact of exponent θ if C=1). It is well known that the complex interpolation space [A⃗]θ is exact of exponent θ. We now turn to geometric interpolation between Hilbert spaces. The best general reference here is McCarthy [7] where more details are given (see also Donoghue [11], Ameur [2] or [21] modulo some evident misprints on page 356, where it should be RanA1/2¯ and RanAθ/2¯, respectively). Let H⃗=(H0,H1) be a regular couple of Hilbert spaces. As is well known, by the Riesz representation theorem (see e.g. [11, p. 253] and [1, p. 261] for more details), there exists a unique, positive self-adjoint operator A in H0 such that ⟨ξ,η⟩H1=⟨A1/2ξ,A1/2η⟩H0,ξ,η∈Δ(H⃗), where DomA1/2=Δ(H⃗) and RanA1/2¯=H0. The operator A is bounded if and only if H0 is contained in H1. Let θ∈(0,1). We define a new inner product on Δ(H⃗) by ⟨ξ,η⟩θ=⟨Aθ/2ξ,Aθ/2η⟩H0,θ∈(0,1). Δ(H⃗) is contained in DomAθ/2 and RanAθ/2¯=H0. The closure of Δ(H⃗), with respect to the norm given by the inner product, we will call Hθ. The space Hθ is a geometric interpolation space of exponent θ and coincides with [H⃗]θ. The proof of the following lemma is partially based on arguments introduced in [21, Lemma 4.1], but for the sake of completeness, all essential details are provided. Lemma 4.1 Let H⃗and K⃗be regular couples of Hilbert spaces. Assume that A and B are positive operators on H0and K0that give the H1and K1inner product, respectively. If T∈L(H⃗,K⃗), then φk(T:Hθ→Kθ)=φk(Bθ/2TA−θ/2:H0→K0)andεk(T:Hθ→Kθ)=εk(Bθ/2TA−θ/2:H0→K0),θ∈[0,1],k∈N. Proof Fix θ∈[0,1]. We denote by HθΔ≔(Δ(H⃗),∥·∥Hθ), H0Δ≔(RanAθ/2,∥·∥H0) and KθΔ≔(Δ(K⃗),∥·∥Kθ), K0Δ≔(RanBθ/2,∥·∥K0). It was concluded in the proof of [21, Lemma 4.1] that the closure of S≔Bθ/2TA−θ/2:H0Δ→K0Δ is bounded as a map between H0 and K0, justifying the later use of Bθ/2TA−θ/2 for the extension of S to H0. Let k∈N. We proceed to show that φk(T:HθΔ→KθΔ)=φk(S:H0Δ→K0Δ). Let 0<ρ<φk(S:H0Δ→K0Δ). By definition, there exists a set of elements x1,…,xn+1∈UH0Δ such that 2ρ<‖S(xi−xj)‖K0Δ,1≤i<j≤n+1. As in the proof of [21, Lemma 4.1], we have ‖T(x˜i−x˜j)‖KθΔ=‖S(xi−xj)‖K0Δ,1≤i<j≤n+1, where x˜1,…,x˜n+1∈UHθΔ. Therefore, φk(S:H0Δ→K0Δ)≤φk(T:HθΔ→KθΔ). In the same manner, we can check the reverse inequality. That εk(T:HθΔ→KθΔ)=εk(S:H0Δ→K0Δ) was already verified in the proof of [21, Lemma 4.1]. An application of Lemma 3.5 completes the proof.□ We can now state our main result. Theorem 4.2 Assume that H⃗=(H0,H1)and K⃗=(K0,K1)are arbitrary couples of Hilbert spaces, T∈L(H⃗,K⃗)and k∈N. Then φk(T:[H⃗]θ→[K⃗]θ)≤64φk(T:H0→K0)1−θφk(T:H1→K1)θandεk(T:[H⃗]θ→[K⃗]θ)≤72εk(T:H0→K0)1−θεk(T:H1→K1)θfor every θ∈(0,1). In the special case θ=1/2, the constants can be improved, namely φk(T:[H⃗]1/2→[K⃗]1/2)≤8φk(T:H0→K0)1/2φk(T:H1→K1)1/2andεk(T:[H⃗]1/2→[K⃗]1/2)≤12εk(T:H0→K0)1/2εk(T:H1→K1)1/2. In the case of operators acting between Hilbert spaces, as it will be in most of this paper, we are interested in improving the estimates obtained (possibly with optimal constants). It seems likely that the arguments employed in the proof of Theorem 4.2 would be much more involved here. Therefore, although these proofs (Theorem 4.2 and [21, Theorem 4.2] modulo some obvious misprints) run along similar lines, there are subtle adjustments necessary to fit the argument to each new situation. Proof Here, therefore, it is assumed for simplicity’s sake that H⃗ and K⃗ are regular. Let A (respectively, B) be the positive operator on H0 (respectively, K0) that gives the H1 (respectively, K1) inner product. Fix θ∈[0,1]. That Bθ/2TA−θ/2∈L(H0,K0) follows from Lemma 4.1. To ease notation, set Rθ=Bθ/2TA−θ/2. Fix k∈N. By Lemma 4.1 again, the kth (inner) entropy number of T from Hθ to Kθ equals the kth (inner) entropy number of Rθ from H0 to K0, namely φk(T:Hθ→Kθ)=φk(Rθ:H0→K0)andεk(T:Hθ→Kθ)=εk(Rθ:H0→K0). (4.1) We first prove a reduced form of the theorem for the family of operators Rθ. Suppose that A−1 and B are bounded. From [21, (3.3)] and Proposition 3.2, we obtain Ek(∣R1/2∣:H0→H0)=Ek2(R1/2*R1/2:H0→H0)1/2≤Ek2(A1/4R1/2*R1/2A−1/4:H0→H0)1/2≤εk(T*:K0→H0)1/2εk(B1/2TA−1/2:H0→K0)1/2. Theorem 3.3 now yields the assertion for θ=1/2 with a constant C=4 or D=6, respectively, φk(R1/2:H0→K0)≤2Cφk(R0:H0→K0)1/2φk(R1:H0→K0)1/2andεk(R1/2:H0→K0)≤Dεk(R0:H0→K0)1/2εk(R1:H0→K0)1/2. (4.2) Now interpolating between R0 and R1/2 or R1/2 and R1 gives the result for θ=1/4 or θ=3/4, respectively. The constant here is equal to (2C)3/2 (respectively, D3/2). Following the same lines, we find that the theorem holds for any dyadic rational in [0,1] with a common constant 4C2 (respectively, D2). Indeed, one may check φk(Rθ:H0→K0)≤4C2φk(R0:H0→K0)1−θφk(R1:H0→K0)θandεk(Rθ:H0→K0)≤D2εk(R0:H0→K0)1−θεk(R1:H0→K0)θ (4.3) for any dyadic rational θ=m/2n∈(0,1) by induction on n. That (4.3) is valid for θ=1/2 is already proved in (4.2). For the inductive step, suppose that (4.3) holds for θ=m/2n, 0<m<2n with a constant equal to D2. It suffices to consider θ=m/2n+1 where 0<m<2n+1 is odd. Now interpolating between R0 and R2θ or R2θ−1 and R1 gives (4.3) for θ<1/2 or θ>1/2, respectively. That ‖Rα−Rβ‖H0→K0→0 as α→β was validated in the proof of [21, Lemma 3.11]. Therefore, our claim is valid for any real θ∈[0,1]. This is because ∣φk(Rα:H0→K0)−φk(Rβ:H0→K0)∣≤‖Rα−Rβ‖H0→K0and∣εk(Rα:H0→K0)−εk(Rβ:H0→K0)∣≤‖Rα−Rβ‖H0→K0 by [21, Proposition 3.5]. We have been working under the assumption that both A−1 and B are bounded. Suppose now that this is no longer so. Take a look at the operators Pn≔∫n−1ndEA−1(λ) and Qn≔∫n−1ndEB(λ), where EA−1 and EB are the corresponding spectral projections. The operators Pn and Qn have bounded extensions on H⃗ and K⃗, which are norm 1 projections on Hθ and Kθ, respectively. Thus QnTPn∈L(H⃗,K⃗). Using equalities (4.1) and Bnθ/2TAn−θ/2=Bθ/2QnTPnA−θ/2, where An−1≔∫n−1nλdEA−1(λ) and Bn≔∫n−1nλdEB(λ) are bounded, and following steps analogous to those above (with A−1, B and Rθ replaced by An−1, Bn, and Bnθ/2TAn−θ/2, respectively), we obtain φk(QnTPn:Hθ→Kθ)≤4C2φk(T:H0→K0)1−θφk(T:H1→K1)θandεk(QnTPn:Hθ→Kθ)≤D2εk(T:H0→K0)1−θεk(T:H1→K1)θ for every θ∈[0,1]. Since Pn (respectively, Qn) are approximate identities on finite subsets of Hθ (respectively, Kθ), it follows that φk(T:Hθ→Kθ)≤4C2φk(T:H0→K0)1−θφk(T:H1→K1)θandεk(T:Hθ→Kθ)≤2liminfn→∞εk(QnTPn:Hθ→Kθ)≤2D2εk(T:H0→K0)1−θεk(T:H1→K1)θ, by Lemma 3.4. The same reasoning applies to (4.2). Since entropy numbers are injective in the sense of [21, Proposition 3.7], the case where the couples H⃗ and K⃗ are not necessarily regular can be handled in much the same way as in the proof of [21, Theorem 4.2].□ Perhaps it is appropriate at this point to note that in order to inductively prove the inequality (4.3) for dyadic fractions, we have to interpolate between one of the spaces H0 or H1, just to keep the constants uniformly bounded (cf. [21, Theorem 4.2]). As an example of the application of Theorem 4.2, we derive the following interpolation estimate in the spirit of Carl’s result. Corollary 4.3 Let H⃗=(H0,H1)be a couple of Hilbert spaces. Then for all T∈L(H⃗), θ∈(0,1)and n∈N, we have En(T:[H⃗]θ→[H⃗]θ)≤En(T:H0→H0)1−θEn(T:H1→H1)θ.In particular, ∣λn(T:[H⃗]θ→[H⃗]θ)∣≤2en(T:H0→H0)1−θen(T:H1→H1)θ. Before going to the proof, it is worth noting that the second statement follows directly from Theorem 4.2 and the Carl–Triebel inequality (see [23, Proposition 3.5] or [5, Theorem 4.2.1]). The proof we give is based instead on the spectral properties of entropy numbers, namely supn∈Nk−1/(2n)(∏i=1n∣λi(T)∣)1/n=limm→∞εkm(Tm)1/m. In a nutshell, various variants of (E) may lead to interpolation results on spectral entropy numbers of this kind, and are therefore of considerable interest. Proof Fix θ∈(0,1). Let k,n,m∈N. Theorem 4.2 now yields εk(Tm:[H⃗]θ→[H⃗]θ)≤Cεk(Tm:H0→H0)1−θεk(Tm:H1→H1)θ. We conclude from Theorem 3.1 that Ek(T:[H⃗]θ→[H⃗]θ)≤Ek(T:H0→H0)1−θEk(T:H1→H1)θ, hence that k−1/(2n)(∏i=1n∣λi(T:[H⃗]θ→[H⃗]θ)∣)1/n≤εk(T:H0→H0)1−θεk(T:H1→H1)θ, and the last assertion follows for k=2n−1.□ Let us remark that in the case of normal operators acting on Hilbert spaces, the analogous interpolation result on entropy numbers follows directly from [21, Theorems 3.8 and 4.3]: Theorem 4.4 Assume that H⃗=(H0,H1)is a regular couple of Hilbert spaces. Let A be a positive operator on H0that gives the H1inner product and let T∈L(H⃗). If the operator T on H0is normal and commutes with A, then 1/8φk(T:H0→H0)≤φk(T:Hθ→Hθ)≤8φk(T:H0→H0)and1/6εk(T:H0→H0)≤εk(T:Hθ→Hθ)≤6εk(T:H0→H0)for every θ∈(0,1]and each k∈N. 5. Conclusion The results of the paper are related to a long standing problem on interpolation of entropy numbers. Edmunds and Netrusov [12] proved that in certain circumstances the entropy numbers of an operator do not behave well under (real) interpolation. Indeed, given any θ∈(0,1) and q∈[1,∞], there are no positive constants α, C such that for all Banach couples (A0,A1), (B0,B1) and every operator T:(A0,A1)→(B0,B1), the inequality εkα(T:A⃗θ,q→B⃗θ,q)≤Cmax{εk(T:A0→B0),εk(T:A1→B1)} holds for every k∈N. However, the (complex) interpolation estimate εk(T:[H⃗]θ→[K⃗]θ)≤72εk(T:H0→K0)1−θεk(T:H1→K1)θ proved directly in this work, is the first one of the type related to the mentioned problem, which holds for all operators between all compatible couples of Hilbert spaces. Funding The author was supported by the Foundation for Polish Science (FNP). References 1 Y. Ameur , A new proof of Donoghue’s interpolation theorem , J. Funct. Spaces Appl 2 ( 2004 ), 253 – 265 . Google Scholar CrossRef Search ADS 2 Y. Ameur , Interpolation of Hilbert spaces, U.U.D.M. Thesis 20 ( 2001 ). 3 J. Bergh and J. Löfström , Interpolation Spaces. An Introduction , Springer-Verlag , Berlin , 1976 , Grundlehren der Mathematischen Wissenschaften, No. 223. 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Teixeira , Interpolation theory and measures of noncompactness , Math. Nachr. 104 ( 1981 ), 129 – 135 . Google Scholar CrossRef Search ADS 15 P. Fernández-Martnez , Interpolation of the measure of non-compactness between quasi-Banach spaces , Rev. Mat. Complut. 19 ( 2006 ), 477 – 498 . 16 S. G. Kren , Yu. I. Petunin and E. M. Semenov , Interpolation of Linear Operators, Translations of Mathematical Monographs Vol. 54 , American Mathematical Society , Providence, R.I. , 1982 . Translated from the Russian by J. Szücs. 17 M. Mastyło and R. Szwedek , Eigenvalues and entropy moduli of operators in interpolation spaces , J. Geom. Anal. 27 ( 2017 ), 1131 – 1177 . Google Scholar CrossRef Search ADS 18 A. Pietsch , Operator Ideals, Mathematische Monographien [Mathematical Monographs] vol. 16 , VEB Deutscher Verlag der Wissenschaften , Berlin , 1978 . 19 R. Szwedek , Measure of non-compactness of operators interpolated by the real method , Studia Math. 175 ( 2006 ), 157 – 174 . 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