SEMINARARBEIT Approximation of Approximation Numbers of Linear Bounded Operators

SEMINARARBEIT

Approximation of Approximation Numbers of Linear Bounded Operators

completed at

Institute for Analysis and Scientic Computing of the Technical University Vienna

supervised by

ao.Univ.Prof. Ph.D. Harald Woracek

Jakob Klein, 1609935 by

Contents

1 Introduction 1

2 Approximation numbers on separable Hilbert spaces 3 2.1 The approximation of approximation numbers on separable Hilbert

spaces . . . 3

3 Approximation under reexivity assumption 7

3.1 The weak and the weak-* topology . . . 7 3.2 The approximation under reexivity assumption . . . 8 3.3 Additional thoughts . . . 12

4 Approximation under duality assumption 15

4.1 The approximation under duality assumption . . . 15 4.2 Additional remarks . . . 18

1 Introduction

In this paper we deal with the approximation of approximation numbers of bounded linear operators on a innite dimensional separable Hilbert space. Thekth approx- imation numbers_kpTq of an operator T describes the distance from T to the subset of all bounded linear operators with an at mostk1-dimensional image. A central fact for this paper is a theorem, that proves the convergence of a sequence s_kpT_nq, n P N to the kth approximation number s_kpTq. The proof of this statement is rather easy and the natural question occurs, if there may be a weakening of the requirements without loosing the results. In this paper we discuss the following two generalized settings:

ˆ First we consider the following situation: Let T : X Ñ Y be a linear and bounded Operator, whereX is a separable andY is a reexive vector space.

ˆ Second we assume that the codomain Y is not reexive but a dual space of a separable normed linear space, Y X¹.

2 Approximation numbers on separable Hilbert spaces

In this chapter we discuss the approximation of the approximation numbers on separable Hilbert spaces. Let B_LpX, Yq denote the set of all bounded linear oper- ators T :X ÑY. Our sources for this chapter are Chapter 1 in [1], Lemma 4.12 in [2] and Chapter 1 in [3].

2.1 The approximation of approximation numbers on separable Hilbert spaces

First we introduce the term of an approximation number:

Denition 2.1 Let X, Y be normed linear spaces. For T P BLpX, Yq and k P N the kth approximation number s_kpTq is dened by

skpTq:inft||T A||:APBLpX, Yq,rankA¤k1u. (2.1) Hence s₁pTq ||T||.

As tAPB_L : rankA ¤k1u „ tA PB_L: rankA¤ku fork P N it follows that s₁pTq ¥s₂pTq ¥...¥0. Let us now formulate the rst version of the Theorem of approximation numbers.

Theorem 2.2 Let H₁, H₂ be separable Hilbert spaces and P_n : H₁ Ñ H₁, Q_n : H₂ Ñ H₂ be linear projections with ||P_n|| ||Q_n|| 1 and @x P H₁ : P_nx Ñ x,

@y P H2 : Qny Ñ y. Let T P BLpH1, H2q and Tn : QnT Pn for all n P N. Then for all k PN,

nlimÑ8s_kpT_nq s_kpTq. (2.2) Before we prove this theorem let us formulate a lemma, that will help us with that:

Lemma 2.3 Let pA_nqnPN „ B_LpH₁, H₂q be a sequence of uniformly bounded Op- erators. For all n P N let rankA_n ¤ k for a k P N. Then there exists an oper- ator A P B_LpH₁, H₂q with rankA ¤ k and for all x P H₁, y P H₂ the sequence py, A_nxqnPN has a convergent subsequence such that lim_lÑ8py, A_nlxq py, Axq.

Proof:

As pA_nqnPN is uniformly bounded, there exists a M ¡ 0 with ||A_n|| ¤ M. As for all n P N rankA_n ¤ k there exist orthonormal sets E_n te^p₁^nq, ..., e^p_k^nqu „ H₁, F_n tf₁^pnq, ..., f_k^pnqu „H₂ and numbers ψ₁^pnq, ..., ψ_k^pnq such that we can express the operatorA_n for all xPH₁ as

A_nx

¸k i1

ψ_i^pnqpx, e^p_i^nqqf_i^pnq. It follows forn PN, that

M ¥ ||A_n|| sup

||x||¤1

||A_nx|| ¥ ||A_ne^p_i^nq||

gf fe¸^k

j1

ψ^p_j^nqpe^p_i^nq, e^p_j^nqqf_j^pnq,

¸k j1

ψ_j^pnqpe^p_i^nq, e^p_j^nqqf_j^pnq c

ψ_i^pnqf_i^pnq, ψ_i^pnqf_i^pnq |ψ^p_i^nq|. (2.3) Let K^H₁¹ : tx P H₁ : ||x|| ¤ 1u, K₁^H2 : tx P H₂ : ||x|| ¤ 1u and K^C_M : tλ P C :|λ| ¤ Mu. We now consider the elements x P H₁ as elements of a dual space x P G¹ H₁ according to the mapping φ : H₁ Ñ G¹, x ÞÑ xˆ with xˆpyq : px, yq.

As a subset of the normed spaceG¹, K^H₁¹ is according to the Theorem of Banach- Alaoglu (see Theorem 5.5.6 in [6]) compact in respect to the weak--Topology. The same way follows, that K₁^H2 is compact with respect to the weak--topology. As K_M^C is compact inC according to the Theorem of Tychono (see Theorem 1.3.1 in [6]) the set

D:

¹k i1

K^C₁

¹k j1

K^H_M¹

¹k l1

K^H_M²

is compact. Therefore the sequence pψ^p₁^nq, ..., ψ^p_k^nq, e^p₁^nq, ..., e^p_k^nq, f₁^pnq, ..., f_k^pnqqnPN has a convergent subsequence, that converges to pψ1, ..., ψk, e1, ..., ek, f1, ..., fkq P D. We now dene

A:H₁ ÑH₂, Apxq:

¸k i1

ψ_ipx, e_iqf_i.

It is clear, that A is bounded. As pe^p_i^nlqqnPN and pf_i^pnlqqnPN are convergent with respect to the weak--topology, it follows that for xPH₁, yP H₂

py, e^p_i^nqq Ñ py, e_iq and px, f_i^pnqq Ñ px, f_iq.

4

Since ψ_i^pnlq Ñψ_i it follows that py, A_nlxq y,°k

i1ψ_i^pnlqpx, e^p_i^nlqqf_i^pnlq

Ñ y,°k

i1ψ_ipx, e_iqf_i

py, Axq.

Now we can prove Theorem 2.2 with help of Lemma 2.3 Proof 2.4 of Theorem 2.2:

First we will show, that lim sup

nÑ8 s_kpT_nq ¤s_kpTq.

As s_kpTq:inft||T A||:AP B_LpH₁, H₂q,rankA¤k1u there exists for every ¡0 an A P B_LpH₁, H₂q with rankA ¤k1 such that ||T A||   s_kpTq . Since for all n PN rankQ_nAP_n¤k1 and ||Q_n|| ||P_n|| 1it follows that

s_kpT_nq ¤ ||T_nQ_nAP_n|| ||Q_nT P_nQ_nAP_n|| ¤ ||Q_n|| ||T A|| ||P_n||

||T A||  skpTq . As ¡0was arbitrary it follows that

lim sup

nÑ8 s_kpT_nq ¤ s_kpTq. (2.4) Now we show lim inf

nÑ8 s_kpT_nq ¥s_kpTq.

Let us assume, that lim inf_nÑ8s_kpT_nq   s_kpTq. As s_kpT_nq ¥ 0 we can assume, thats_kpTq ¡0. Hence there has to exist aδP p0, s_kpTqq such that for everyn₀ PN there exists a n PN with n¥n₀, such that

s_kpT_nq ¤s_kpTq δ.

According to the denition of skpTnq for every P p0, skpTq δq we can nd a A PB_LpH₁, H₂q withrankA ¤k1 such that

||T_nA|| ||P_nT P_nA||  s_kpTq δ .

Now according to Lemma 2.3 there exists an Operator AP B_LpHq with rankA ¤ k 1 such that A is a limit of a subsequence of pAqPp0,skpTqδq that enjoys the following property

@xPH₁@yPH₂ lim

jÑ0py, A_jxq py, Axq. (2.5) LetxPH1, yPH2 such that ||x|| ||y|| 1. Then follows that

|py, Q_nT P_nxq py, Axq| ¤ ||Q_nT P_nA||  s_kpTq δ . As the inner product of H₂ is continuous, this implies, that

|py, T xq py, Axq| lim_jÑ0|py, T_nxq py, A_jxq| ¤s_kpTq δ .

Hence ||T A|| ¤skpTq δ and therefore skpTq ¤skpTq δ  skpTq. This is a contradiction, so the assumptionlim inf_nÑ8s_kpT_nq  s_kpTq was wrong. Hence

lim inf

nÑ8 s_kpT_nq ¥ s_kpTq. (2.6) Together with equation (2.4) it follows that

@k PN: lim

nÑ8s_kpT_nq s_kpTq.

6

3 Approximation under reexivity assumption

As we now got a rst answer of how we can approximate the approximation num- bers of a linear bounded operator, we now want to evolve these results to a more general setting. Instead of Hilbert spaces we want to consider normed spacesX, Y and a linear bounded mapping T : X ÑY. In addition we want to have weaker requirements to the sequence pT_nqnPN „ B_LpX, Yq that converges to T for ap- proximating s_kpTq. As we will see, we need to assume one more condition to the codomain of the OperatorT. In this chapter we will assume, thatX is a separable and Y is a reexive normed space. First we will introduce terms of topology, in order to formulate the nal theorem of this chapter. Second we will prove some lemmas, that will help us for our further conclusions and nally we will prove the main theorem of this chapter.

3.1 The weak and the weak-* topology

We now will introduce the terms of the weak and the weak--topology. This section follows the thoughts of Chapter 5.3 in [6].

Denition 3.1 Let X be a vector space and Y be a point separating linear sub- space of the algebraic Dual space X. The weak topology with respect to Y onX is dened as the initial topology with respect to all functions y P Y. This means, that the weak topology on X is the coarsest topology such that all y P Y are con- tinuous. The weak topology on X denotes the weak topology on X with respect to the topological dual space X¹.

Denition 3.2 Let X and Y are normed spaces. We consider the set Z :spantT ÞÑφpTpxqq:xPX, φ PY¹u.

It is clear, that Z is a point separating subspace of the algebraic Dual space of B_LpX, Yq. The weak topology on B_LpX, Yq with respect to Z is called the weak operator topology and is denoted by T_ω.

Remark 3.3 Hence for a sequence pT_nqnPN€B_LpX, Yq holds that

pT_nqnPNÑT in T_ω ô @xP X@φPY :φpT_nxq ÑφpT xq.

LetX be a vector space and Y be a subspace ofX. Let us consider the mapping ι : X Ñ Y with ιpxq: Y Ñ C, ιpxqpyq :ypxq. Then ιpXq is a point separating subspace of Y. Therefore the following denition is well dened:

Denition 3.4 Let X be a vector space and ι be dened as in Remark 3.3. The weak topology onX¹ with respect to ιpXq is called the weak-*-topology onX¹ and is denoted by T_ω.

Remark 3.5 Let X and Y be normed linear spaces. Let pT_nqnPN be a sequence with T_n P B_LpX, Y¹q. The sequence pT_nqnPN converges to T with respect to the weak- operator topology if

@xPX@yPY :T_nxpyq ÑT xpyq as nÑ 8.

It follows that strong operator convergence implies weak operator convergence and weak operator convergence implies weak- operator convergence.

3.2 The approximation under reexivity assumption

Now we will formulate the theorem of approximation under the reexivity assump- tion. Before we prove this theorem, we will discuss two lemmas, that will help us with that. This section follows Chapter 2 of [3].

Theorem 3.6 Let X be a separable normed linear space and Y be a reexive Banach space. In addition let T P B_LpX, Yq and pP_nqnPN € B_LpXq, pQ_nqnPN € B_LpYq be sequences of operators with @n P N ||P_n|| ¤ 1,||Q_n|| ¤ 1. We dene T_n:Q_nT P_n. If T_nÑT in T_ω as nÑ 8, then holds that

@kP N lim

nÑ8s_kpT_nq s_kpTq.

The rst of the two lemmas is the following:

Lemma 3.7 LetX, Y be normed linear spaces and T P B_LpX, Yq with rankT k for a k PN. Then holds that there exists a basis tb₁, ..., b_ku of TpXq „ Y and a set tφ₁, ..., φ_ku € X¹ with @j P t1, ..., ku:||b_j|| 1 and ||φ_j|| ¤ ||T||, such that

@xP X :T x

¸k j1

φ_jpxqb_j.

8

Proof:

As TpXq is the image of a linear space under a linear mapping with rankk we can nd a basis ta₁, ..., a_ku of TpXq with @j :||a_j|| 1. Let K^T₁^pXq: tyPTpXq:

||y|| ¤1u. As K₁^TpXq is closed, it follows that

K :

¹k i1

K^T₁^pXq

is closed. Let us consider the mapping det :K ÑC. According to linear algebra we know, that this mapping is continuous and

detpx₁, ..., x_kq ¡0ô px₁, ..., x_kq is linear independent.

Since K is closed and det is continuous there exists a maximum in K. Let this maximum be denoted by pb1, ..., bkq. As pb1, ..., bkq PK it holds that @j P t1, ..., ku:

||b_j|| ¤ 1. Being the maximum on K in follows that detpb₁, ..., b_kq ¥ 1, because detpa₁, ..., a_kq 1. In particular follows, that all b_j 0. It holds that

@j P t1, ..., ku: detpb₁, ..., b_kq ¥detpb₁, ...,_||b^bj

j||, ..., b_kq ^detpb_||b^1,...,b_j|| ^kq and therefore @j P t1, ..., ku:||b_j|| ¥ 1. Together follows, that

@j P t1, ..., ku:||b_j|| 1. (3.1) Let for alli U_i :spanptb₁, ..., b_kuztb_iuq. Thenb_i R U_iand we see, thatdistpb_i, U_iq ¤ 1 and for every uPU_i, that ||b_iu|| ¡0. Considering

1detpb1, ..., bkq ¥detpb1, ..., bi1,_||^b_b^iu

iu||, bi 1, ..., bkq ^det_||^p_b^b_i^1,...,b_u||^kq

we see, that ||biu|| ¥ 1for alluPUi. In conclusion follows, thatdistpbi, Uiq 1. Let us for j P t1, ..., ku dene f_j : spantb_ju Ñ C, f_jpγb_jq:γ. Then follows, that

||f_j|| 1. According to the Theorem of Hahn-Banach (see Theorem 5.2.3 in [6]) there exist @j P t1, ..., ku an extension Fj : X ÑC, Fj|spantbju fj with Fj P X¹,

||F_j|| 1 and F_j|Uj 0. Since F_j is an extension off_j it follows that

@i, j P t1, ..., ku:F_jpb_iq δ_ij.

As pb1, ..., bkq is a basis of TpXq we can describe every xPTpXq as x°k i1γibi. In conclusion it holds, that

@yPTpXq:y

¸k j1

F_jpyqb_j.

Let us now for allj P t1, ..., ku dene

φ_j :X ÑC, φ_jpxq:F_jpT xq. (3.2) Then φ_j PX¹ and it follows, that ||φ_j|| ||F_j T|| ¤ ||F_j|| ||T|| ||T|| and

T x

¸k j1

φjpxqbj. (3.3)

Let us now formulate a second lemma, that corresponds to Lemma 2.3 before we will start with the proof of Theorem 3.6 (see Lemma 2.4 in [3]).

Lemma 3.8 Let X be a separable normed linear space, Y be a reexive Banach space and pA_nqnPN € B_LpX, Yq a sequence of uniformly bounded operators with

@nPN: rankpA_nq ¤ k. Then there exists an operatorAP B_LpX, Yq withrankA¤ k and a subsequence pA_njqjPN such that

A_nj ÑA in T_ω Proof:

Since pA_nqnPN is uniformly bounded, there exists a M ¡0 with @n PN :||A_n|| ¤ M. As for alln PN:AnPBLpX, Yq and rankAn¤k by a consequence of Lemma 2.3 there existb^p₁^nq, ..., b^p_k^nqP A_npXq and φ^p₁^nq, ..., φ^p_k^nq PX¹ with

@n PN@iP t1, ..., ku:||b^p_i^nq|| 1 and ||φ^p_i^nq|| ¤ ||A_n|| ¤ M such that

@xP X:Anx

¸k i1

φ^p_i^nqpxqb^p_i^nq.

According to the Theorem Banach-Alaoglu (see Theorem 5.5.6 in [6]) for every i there exists a subsequence pφ^p_i^njqqnjPN, that converges in T_ω to φ_i P X¹. Since Y is a reexive Banach space by consequence of the Eberlein-Smulian Theorem for every i there exists a subsequence pb^p_i^njqqnjPN, that converges in T_ω to b_i P Y (see Theorem 8.25 in [5]). Let us dene

A:X ÑY,@xPX :Ax

¸k i1

φ_ipxqb_i. (3.4) It is clear, thatrankA¤k and A PB_LpX, Yq. Hence @f PY¹ holds that

jlimÑ8fpA_njxq lim

jÑ8

¸k i1

φ^p_i^njqpxqfpb^p_i^njqq

¸k i1

φ_ipxqfpb_iq fpAxq.

10

Let us now prove Theorem 3.6 with help of Lemma 3.8. This proof is inspired of theorem 2.8 in [3]. The main idea is similar to the proof of Theorem 2.2, that corresponds with Theorem 3.6.

Proof of Theorem 3.6:

As in proof 2.4 we will rst show, that lim sup_nÑ8s_kpT_nq ¤ s_kpTq and then with a contradiction, that lim inf_nÑ8s_kpT_nq ¥s_kpTq.

Similar to proof 2.4 for all k P N and every ¡ 0 there exists an A P B_LpX, Yq with rankA ¤k1and ||T A||  s_kpTq . Then for all nP Nholds that s_kpT_nq ¤ ||Q_nT P_nQ_nAP_n|| ¤ ||Q_n|| ||T A|| ||P_n|| ¤ ||T A||   s_kpTq . Since was arbitrary, we get

lim sup

nÑ8 s_kpT_nq ¤ s_kpTq. (3.5) Now we proof the other inequality. The conclusion holds if s_kpTq 0. Again as in proof 2.4 we now assume, that s_kpTq ¡ 0 and liminf_nÑ8s_kpT_nq   s_kpTq.

Therefore it follows, that

DP p0, s_kpTqq@nPNDn_l¥n:s_kpT_nlq  s_kpTq .

Hence for every l P N there exists an A_nl P B_LpX, Yq with rankA_nl ¤ k1 such

that ||T_nlA_nl||   s_kpTq . (3.6)

Therefore holds that

||A_nl|| ¤ ||A_nlT_nl|| ||T_nl||   s_kpTq ||T||.

Thus the sequence pA_nlqlPN is uniformly bounded and the assumptions of Lemma 3.8 are fullled. Therefore there exists a subsequence pA_jqjPN „ pA_nlqlPN and an operator A :X ÑY such that rankA ¤k1 and A_j ÑA inT_ω as j Ñ 8. Let x P X and f P Y¹ be arbitrary with ||x|| ¤ 1 and ||f|| ¤ 1. We want to get the following term small:

@j P N:|fpT xqfpAxq| ¤ |fpT xqfpT_jxq| |fpT_jxqfpA_jxq| |fpA_jxqfpAxq|. (3.7) First we see, that

|fpT_jxq fpA_jxq| ¤ ||f|| ||T_jA_j||   s_kpTq .

As pA_jqjPNÑA it follows that there exists a j₀ P N such that for allj ¡j₀

|fpA_jxq fpAxq|   ₃.

Since pT_nqnPN ÑT inT_ω, there exists a j₁ PN such that for all j ¡j₁

|fpT xq fpTjxq|   ₃.

In conclusion follows, as x P X with ||x|| ¤ 1 and f P Y¹ with ||y|| ¤ 1 were arbitrary

s_kpTq ¤ ||T A|| ¤s_kpTq ₃. This is a contradiction, hence

lim inf

nÑ8 s_kpT_nq ¥ s_kpTq. (3.8) Together with (3.5) it follows, that

nlimÑ8s_kpT_nq s_kpTq.

Remark 3.9 According to Eberlein's Theorem (see Theorem 16.5 in [4]) and propo- sition 2.7 in [3] ifY is a non-reexive space, then there has not to be a subsequence of pT_nqnPN, that is convergent in the weak operator topology. Therefore the demand, that Y is a reexive space, is necessary.

3.3 Additional thoughts

In this section we want to discuss whether the approximation number of an oper- ator s_kpTq is a minimum or not. At the very end of this chapter we add one last version of the approximation under reexivity assumption. This section follows 2.5 and 2.9 in [3].

Corollary 3.10 LetXbe a separable normed linear space ,Y be a reexive Banach space and T P B_LpX, Yq, k P N. Then there exists an operator A P B_LpX, Yq, rankA¤k1 with

s_kpTq ||T A||

Proof:

According to the denition ofs_kpTq for everyn PNthere exists anA_n PB_LpX, Yq with rankA_n ¤k1 and

||T A_n|| ¤ s_kpTq 1

n (3.9)

12

and therefore ||A_n|| ¤ ||T|| s_kpTq _n¹   ||T|| s_kpTq 1. As pA_nqnPNis uniformly bounded by consequence of Lemma 3.8 there exists an A P B_LpX, Yq, rankA ¤ k1and a subsequence pA_nlqlPN with

pA_nlqlPNÑA inT_ω.

Let ¡0, xPX with ||x|| ¤1 and f PY¹ with ||f|| ¤1be arbitrary. Let n PN be such that for all n_l ¥n holds

1

nl   ₂ and |fpAxq fpA_nlxq|   ₂. Together it follows that

|fpT xq fpAxq| ¤ |fpT xq fpA_nlxq| |fpA_nlxq fpAxq| ¤

||f|| ||x|| ||T A_nl|| ₂  s_kpTq _n¹ ₂  s_kpTq . As x, f and were arbitrary, it follows

||T A|| ¤s_kpTq. (3.10) By consequence of the denition of skpTq follows, that ||T A|| ¥ skpTq and together with (3.10) we get

||T A|| s_kpTq.

Last we add a corollary, that gives one more version of the approximation of the approximation numbers under reexivity assumption.

Corollary 3.11 LetX be a separable normed linear space ,Y be a reexive Banach space and T PB_LpX, Yq, k PN. Let pP_nqnPN„B_LpXq and pQ_nqnPN „B_LpYq such that ||P_n|| ¤ 1 with @xP X :P_nx Ñ x and ||Q_n|| ¤ 1 with @y P Y : Q_ny Ñ y in T_ω. Then holds that

@kP N:s_kpT_nq Ñs_kpTq for nÑ 8.

Proof:

We have to show, that T_n:Q_nT P_n ÑT in T_ω. Therefore let x P X and f P Y¹ be arbitrary. As @xPX :P_nxÑx and @yP Y :Q_nyÑy inT_ω follows

|fpTnxq fpT xq| ¤ |fpQnT Pnxq fpQnT xq| |fpQnT xq fpT xq| ¤

||f|| ||Q_n|| ||T|| ||P_nxx|| |fpQ_nT xq fpT xq| Ñ 0as nÑ 8.

Hence T_nÑT in T_ω and therefore the conclusion follows from Theorem 3.6.

4 Approximation under duality assumption

In the last chapter we discussed the approximation of approximation numbers of a linear bounded operator T : X Ñ Y on normed spaces under the assumption, that Y is a reexive space. We now want to show a second possibility to proof the approximation on normed spaces under slightly dierent conditions. The main dierence will be, that we assume, that the codomain Y of T is not necessarily reexive but the dual space of a separable normed linear space. In addition we will slightly change the assumptions on the sequence of operators pT_nqnPN. In corollary 3.11, our nal version of the approximation in the last chapter, we asked for the two sequences pP_nqnPN, pQ_nqnPNfor pP_nqnPNpoint wise convergence and for pQ_nqnPN

convergence in the weak operator topology. Now we even want to weaken that, namely that T_n :Q_nT P_n is just convergent in the weak-* operator topology. In the rst section of this chapter we will rst prove a lemma, similar to Lemma 3.8, that will help us to prove the main theorem of this chapter. That we will do right after discussing this lemma. In the second section of this chapter we will discuss, if the approximation number of an operator s_kpTq is a minimum or not under the duality assumption. Finally, we will give one very last version of the approximation and sum up all three versions we proofed.

4.1 The approximation under duality assumption

Let us now formulate a lemma, that corresponds to Lemma 3.8. This section follows Lemma 3.1 and Theorem 3.3 in [3].

Lemma 4.1 Let X, Z be separable normed linear spaces and Y : Z¹, k P N.

Let pAnqnPN „ BLpX, Yq be a sequence of uniformly bounded operators such that

@n P N : rankA_n ¤ k. Then there exists a subsequence pA_nlqlPN and an operator AP B_LpX, Yq with rankA¤k such that

A_nl ÑA in T_ω as n Ñ 8.

Proof:

By consequence of Lemma 3.7 for every n P N there exist a basis pb^p₁^nq, ..., b^p_k^nqq

of ApXq „ Y with @j P t1, ..., ku : ||b_j|| 1 and pφ^p₁^nq, ..., φ^p_k^nqq „ X¹ with @j P t1, ..., ku:||φ^p_j^nq|| ¤ ||T|| such that

@xPX :A_nx

¸k j1

φ^p_j^nqpxqb^p_j^nq.

Since X¹ and Y are dual spaces of separable normed spaces, according to the Theorems of Banach-Alaoglu and Tychono (see Theorem 5.5.6 and Theorem 1.3.1 in [6]), the sequences pb^p₁^nq, ..., b^p_k^nqqnPNand pφ^p₁^nq, ..., φ^p_k^nqqnPNhave weak-* convergent subsequences. Hence there exist pb₁, ..., b_kq and pφ₁, ..., φ_kq such that

@xPX :pφ^p₁^nlqpxq, ..., φ^p_k^nlqpxqq Ñ pφ₁pxq, ..., φ_kpxqq and

@z PZ :pb^p₁^nlqpzq, ..., b^p_k^nlqpzqq Ñ pb₁pzq, ..., b_kpzqq as nÑ 8.

let us now dene, as in Lemma 3.8

@xPX :Ax:

¸k j1

φ_jpxqb_j.

It is clear, that AP B_LpX, Yq with rankA¤k. Hence for every xPX and z PZ follows, that

lim

lÑ8A_nlxpzq lim

lÑ8

¸k j1

φ^p_j^nlqpxqb^p_j^nlqpzq

¸k j1

φ_jpxqb_jpzq Axpzq.

Let us now formulate the main theorem of this chapter. It proves the approxima- tion of the approximation numbers under the duality assumption.

Theorem 4.2 LetX, Z be separable normed linear spaces andY :Z¹,k PN. Let T P B_LpX, Yq and pP_nqnPN „B_LpXq, pQ_nqnPN „B_LpYq be sequences of operators with @n PN:||P_n|| ¤ 1, ||Q_n|| ¤1 such thatT_n:Q_nT P_nÑT in T_ω as nÑ 8.

Then holds

nlimÑ8s_kpT_nq s_kpTq.

Proof:

As in the proof of Theorem 3.3 we again show rstlim sup_nÑ8s_kpT_nq ¤s_kpTq and then lim inf_nÑ8s_kpT_nq ¥ s_kpTq. The rst inequality can be absolutely similarly shown as in Theorem 3.3. So we have

lim sup

nÑ8 s_kpT_nq ¤ s_kpTq. (4.1)

16

Again we want to provelim inf_nÑ8s_kpT_nq ¥s_kpTq by contradiction. So we assume, that lim inf_nÑ8s_kpT_nq   s_kpTq. In consequence there exist an ¡0 and innitely many n_l PN with s_kpT_nlq   s_kpTq . Therefore we nd a sequence of operators pA_nlqnPN „ B_LpX, Yq with @l P N : rankA_nl ¤ k 1 such that ||A_nl T_nl||  

s_kpTq . It follows, that

@lP N:||A_nl|| ¤ ||A_nlT_nl|| ||T_nl||   s_kpTq ||T||.

As pA_nlqnPN is uniformly bounded by consequence of Lemma 4.1 there exist a subsequence pAjqjPN „ pAn_lqlPN and an operator APBLpX, Yq such that

pAjqjPNÑA inTω.

Let x P X and z P Z with ||x|| ¤ 1 and ||z|| ¤ 1. We want to get the following term small:

|T xpzq Axpzq| ¤ |T xpzq T_jxpzq| |T_jxpzq A_jxpzq| |A_jxpzq Axpzq|. (4.2) Now rst we see, that

@j PN:|T_jxpzq A_jxpzq| ¤ ||T_j A_j||  s_kpTq .

Second, by weak-*-convergence of pT_jqjPN and pA_jqjPN there exists a j such that

|T xpzq T_jxpzq|   ₃ and

|Axpzq A_jxpzq|   ₃.

Hence |T xpzq Axpzq|   s_kpTq ₃. As x P X and z P Z with ||x|| ¤ 1 and

||z|| ¤1 were arbitrary it follows that

s_kpTq ¤ ||T A|| ¤ s_kpTq ₃. We have reached a contradiction. Hence

lim inf

nÑ8 s_kpT_nq ¥s_kpTq. (4.3) Together with (4.1) we have

nlimÑ8skpTnq skpTq.

4.2 Additional remarks

Similar to section 3.3 we now want to show, that under the duality assumption the approximation numbers are minima, so for every k P N there exists an operator A : X Ñ Y such that s_kpTq ||T A||. Last we give one nal version of the approximation of the approximation numbers under the duality assumption. This section follows Corollary 3.2 and Corollary 3.4 in [3].

Corollary 4.3 Let X, Z be separable normed linear spaces and Y : Z¹, k P N.

LetT P B_LpX, Yq. Then there exists an operatorAP B_LpX, Yq withrankA¤k1 such that

s_kpTq ||T A||.

Proof:

Similar to the proof of Corollary 3.10, according to the denition ofs_kpTq for every nPN there exists an A_n PB_LpX, Yq withrankA_n ¤k1and

||T A_n|| ¤ s_kpTq 1

n (4.4)

and therefore ||A_n|| ¤ ||T|| s_kpTq _n¹   ||T|| s_kpTq 1. As pA_nqnPNis uniformly bounded by consequence of Lemma 4.1 there exists an A P BLpX, Yq, rankA ¤ k1 and a subsequence pA_nlqnPN with

pA_nlqnPN ÑA in T_ω.

Let ¡ 0, x P X, z P Z with ||x|| ¤1 and ||z|| ¤ 1. Letn P N be such that for alln_l¥n holds that

1

n_{l   2} and |Axpzq A_nlxpzq|   ₂. Together follows that

|T xpzq Axpzq| ¤ |T xpzq A_nlxpzq| |A_nlxpzq Axpzq| ¤

||x|| ||z|| ||T A_nl|| ₂  s_kpTq _n¹ ₂  s_kpTq . As x, z and were arbitrary, it follows

||T A|| ¤skpTq. (4.5)

By consequence of the denition of s_kpTq follows, that ||T A|| ¥ s_kpTq and together with (4.5) we get

||T A|| s_kpTq.

18

Last we name a corollary, that gives one very last version of the approximation of the approximation numbers under duality assumption.

Corollary 4.4 Let X, Z be separable normed linear spaces and Y : Z¹, k P N. Let T P BLpX, Yq and pPnqnPN „ BLpXq, pQnqnPN „ BLpYq be sequences of operators with @n PN:||P_n|| ¤ 1 such that @xP X :P_nxÑx and ||Q_n|| ¤1 with

@yP Y :Q_nyÑy in T_ω as nÑ 8. Then holds that

nÑ8lim skpTnq skpTq.

Proof:

We want to use Theorem 4.2. Therefore we have to show, that Tn :QnT Pn ÑT in T_ω as nÑ 8. As @xPX :P_nxÑx and @yPY :Q_nyÑy inT_ω it follows

|T_nxpzq T xpzq| ¤ |Q_nT P_nxpzq Q_nT xpzq| |Q_nT xpzq T xpzq| ¤

||Qn|| ||T|| ||Pnxx|| ||z|| |QnT xpzq T xpzq| Ñ0 asn Ñ 8.

Hence T_nÑT in T_ω and therefore the conclusion follows from Theorem 4.2.

4.3 Conclusion

Let us now formulate a theorem that sums up all three versions of the approxima- tion of the approximation numbers.

Theorem 4.5 (Approximation of approximation numbers) Let X and Y be normed spaces andT PB_LpX, Yq. Furthermore let pP_nqnPN„B_LpXq, pQ_nqnPN „ B_LpYq be sequences of Operators with ||P_n|| ¤ 1, ||Q_n|| ¤ 1 for all n P N. Let

@n PNT_n:Q_nT P_n. If one of the following statements

ˆ X andY are Hilbertspaces. P_n and Q_n are linear projections with P_nÑid_X and Q_nÑid_Y.

ˆ X is a separable, Y a reexive Banachspace and T_nÑT in T_ω.

ˆ X and Z are separable normed spaces, Y :Z¹ and T_n ÑT in T_ω. is true, then holds that

nlimÑ8skpTnq skpTq.

Bibliography

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[3] K. P. Deepesh, S. H. Kulkarni, and M. T. Nair. Approximation numbers of operators on normed linear spaces. 2009.

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