Essential Approximate Point Spectra for Upper Triangular Matrix of Linear Relations

Acta Mathematica Scientia 2013,33B(4):1187–1201 http://actams.wipm.ac.cn

ESSENTIAL APPROXIMATE POINT SPECTRA FOR UPPER TRIANGULAR MATRIX OF LINEAR RELATIONS∗ Souhir ELLEUCH

Maher MNIF

D´ epartement de Math´ ematiques, Universit´ e de Sfax, Facult´ e des Sciences de Sfax, Laboratoire de Physiques Math´ ematiques, B.P. 1171, 3000 Sfax, Tunisie E-mail: elleuch− [email protected]; [email protected]

Abstract When A ∈ LR(H) and B ∈ LR(K) are given, for C ∈ LR(K, H) we denote by MC the linear relation acting on the infinite dimensional separable Hilbert space H ⊕ K   A C of the form MC = . In this paper, we give the necessary and sufficient conditions 0 B on A and B for which MC is upper semi-Fredholm with negative index or Weyl for some C ∈ LR(K, H). Key words linear relation; Fredholm and semi-Fredholm relation; essential approximate point spectra for linear relation 2010 MR Subject Classification

1

47A06; 47A10; 47A53; 47A55

Introduction

Throughout this paper, H and K are infinite dimensional separable Hilbert spaces over K = R or C. A linear relation T : H → K is a mapping from a subspace D(T ) = {u ∈ H : T (u) 6= ∅} ⊆ H, called the domain of T, and taking values in P (K) \ {∅} (the collection of nonempty subsets of K) such that T (αx1 + βx2 ) = αT (x1 ) + βT (x2 ) for all non zero scalars α, β ∈ K and x1 , x2 ∈ D(T ). We denote by LR(H, K) the class of linear relations everywhere defined and we write LR(H) := LR(H, H). If T maps the points of its domain to singletons, then T is said to be an operator, that’s equivalent to T (0) = {0}. The class of linear bounded operators is denoted by L(H, K). The graph G(T ) of T is G(T ) = {(u, v) ∈ H × K : u ∈ D(T ), v ∈ T (u)}. The closure of T, denoted by T , is the linear relation defined by G(T ) := G(T ). ∗ Received

June 9, 2011; revised February 14, 2012.

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The inverse of T is the relation T −1 given by G(T −1 ) = {(v, u) ∈ K × H : (u, v) ∈ G(T )}. If G(T ) is closed, then T is said to be closed. The class of such relations is denoted by CR(H, K). We design by R(T ) = T (D(T )) the range of T and N (T ) := {x ∈ H : (x, 0) ∈ G(T )}. If R(T ) = K, then T is called surjective and if N (T ) = {0}, then T is called injective. Notice that when u ∈ D(T ), v ∈ T (u) ⇔ T (u) = v + T (0). We write n(T ) = dim N (T ) and d(T ) = codimR(T ) and the index of T, i(T ), is defined by i(T ) = n(T ) − d(T ) provided n(T ) and d(T ) are not both infinite. We denote by Rφ+ (H, K) = {T ∈ CR(H, K); R(T ) is closed and n(T ) < +∞}; the class of upper semi-Fredholm relations, Rφ− (H, K) = {T ∈ CR(H, K); R(T ) is closed and d(T ) < +∞}; the class of lower semi-Fredholm relations and Rφ− + (H, K) = {T ∈ Rφ+ (H, K); i(T ) ≤ 0}. T is called Fredholm relation, if T ∈ Rφ+ (H, K) ∩ Rφ− (H, K). The class of all Fredholm relations is denoted by Rφ(H, K). If, in addition, i(T ) = 0, then T is said Weyl relation. Let E be a closed subset of H. We denote by QH E ∈ L(H, H/E) the natural quotient map from H onto H/E. If T ∈ LR(H, K), then we denote the quotient QK ∈ L(K, K/T (0)) by T (0) QT . It is easy to see that QT T is single valued. For u ∈ D(T ), we define kT uk := kQT T uk, i.e., kT uk = d(v, T (0)) for all v ∈ T (u). The norm of T is defined by kT k := kQT T k, and thus kT k =

sup

kT uk. We note that this quantity is not a true norm since kT k = 0

u∈BD(T )

does not imply T = 0. Notice that for u ∈ D(T ), QT T (u) = QT (v) for all v ∈ T (u). Indeed, since v ∈ T (u) if and only if T (u) = v + T (0) then QT T (u) = QT (v). A relation T ∈ LR(H, K) is said to be continuous if for any neighborhood V ⊂ R(T ), the inverse image T −1 (V ) is a neighborhood in H. It can be shown that T is continuous if and only if kT k < +∞ (see Cross [1, II.3.2]). If T is an everywhere defined relation in LR(H, K)

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such that kT k < +∞ then T is said to be bounded. The class of such relations is denoted by BR(H, K). The class of all closed bounded relations and semi-Fredhoml bounded relations will be denoted by CBR(H, K) and BRφ± (H, K), respectively. If T ∈ CR(H), we define σSF+ (T ) = {λ ∈ C; (T − λ I) is not in Rφ+ (H)} called the upper semi-Fredholm spectra of T ; σSF− (T ) = {λ ∈ C; (T − λ I) is not in Rφ− (H)} called the lower semi-Fredholm spectra of T ;  σea (T ) = λ ∈ C; (T − λ I) is not in Rφ− + (H) called the essential approximate point spectra of T. The minimum modulus of a linear relation T is the quantity    if D(T ) ⊂ N (T );  +∞,   γ(T ) = kQT T xk   inf : x ∈ D(T )\N (T ) , otherwise.  dist (x, N (T ))

Notice that R(T ) is closed if and only if γ(T ) > 0 (see Cross [1, II.3.2 and III.4.2]). Let T be a linear relation in LR(H, K). Then the adjoint relation T ∗ ∈ LR(K, H) is defined by (see [3]): G(T ∗ ) = {(v, v ′ ) ∈ K × H : hu′ , vi = hu, v ′ i for all (u, u′ ) ∈ G(T )}. For T ∈ LR(H, K), we have those equalities N (T ∗ ) = R(T )⊥ ;

T ∗ (0) = D(T )⊥ ;

N (T ) = R(T ∗ )⊥ ;

T (0) = D(T ∗ )⊥

(see Cross [1, III.1.4]). So, we have if D(T ) = H, then T ∗ is an operator. It is easy to see that if T ∈ CR(H, K) with closed range, then n(T ∗ ) = d(T ) and d(T ∗ ) = n(T ) and hence T ∈ Rφ+ (H, K) if and only if T ∗ ∈ Rφ− (K, H) and i(T ∗ ) = −i(T ). Let A ∈ LR(H) and B ∈ LR(K). For a given C ∈ LR(K, H), we denote by   A C MC = 0 B the linear relation acting on H ⊕ K defined by      u1 v1 G(MC ) = , ∈ (H × K)2 ; v1 ∈ Au1 + Cu2 and v2 ∈ Bu2 . u2 v2 The purpose of this work is to extend some properties of matrix operators developed in [4] to a large class that is the class of linear relations. Essentially, in this paper, it is shown that a

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2 × 2 relation matrix MC is upper semi-Fredholm and i(MC ) ≤ 0 for some C ∈ L(K, H) if and only if A is upper semi-Fredholm and     • n(B) < +∞ and n(A) + n(B) ≤ d(A) + d(B) (1.1) or n(B) = d(A) = +∞, if R(B) is closed,    • d(A) = +∞, if R(B) is not closed.

We also give main necessary and sufficient conditions for which MC is Weyl or MC is lower semi-Fredholm with nonnegative index for some C ∈ L(K, H). The outline of this paper is as follows. In Section 2, we characterize QMC MC and the adjoint of MC which will be very useful in the sequel. In Section 3, we expresses the essential approximate point spectrum of MC with that of M0 . In Section 4, we generalize the results given for 2 × 2 matrix relation to a n × n upper triangular matrix relation with n ≥ 2.

2

Preliminaries Results

This section contains some definitions and auxiliary results which will be needed in the rest of this paper. Proposition 2.1 Let A ∈ LR(H), B ∈ LR(K) and C ∈ LR(K, H) such that C(0) ⊆ A(0). Then we have   QMC MC = 

QA A QA C

. QB B       x u x , i.e., there exist u1 ∈ Ax and u2 ∈ Cy ∈ MC ∈ H × K and Proof Let y v y such that u = u1 + u2 ∈ Ax + Cy and v ∈ By. This implies that     u x . = QMC QMC MC v y   u . Now, let find the expression of QMC       v   u u e u u e ∈ MC (0), i.e., − if and only if ∈ QMC Notice that v ve v ve  u e − u ∈ A(0) + C(0) = A(0), (2.1)  ve − v ∈ B(0).

That’s equivalent to

0

 u e ∈ QA (u1 + u2 ),  ve ∈ QB (v).

(2.2)

    u QA u 1 + QA u 2 This shows that QMC = . v QB v Since u1 ∈ A(x), u2 ∈ Cy and v ∈ By, it follows that QA u1 = QA Ax, QA u2 = QA Cy and QB v = QB By.

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As a conclusion,        x QA Ax + QA Cy QA A QA C x =  . QMC MC   =  y QB By 0 QB B y This achieves the proof. 2 Proposition 2.2 Let A ∈ LR(H), B ∈ LR(K) and C ∈ LR(K, H). Then the adjoint of MC is the single valued relation acting on H ⊕ K of the form   ∗ A 0 . MC∗ =  C ∗ B∗ Proof Let

    u x and ∈ H × K. By the definition of the adjoint relation on Hilbert v y

spaces we have         ′  u x u x ∗ MC , = , = hu, x′ i + hv, y ′ i. v y v y′ Since x′ ∈ Ax + Cy, then there exist x1 ∈ Ax and x2 ∈ Cy such that x′ = x1 + x2 . Then hu, x′ i + hv, y ′ i = hu, x1 i + hu, x2 i + hv, y ′ i = hA∗ u, xi + hC ∗ u, yi + hB ∗ v, yi = hA∗ u, xi + hC ∗ u + B ∗ v, yi       ∗ x u A 0 . , = ∗ ∗ y v C B This completes the proof. 2 Lemma 2.1 (Cross [1, III.1.10]) Let U be a subset of H and let JU denote the natural injection map of U into H, i.e., D(JU ) = U and JU x = x for all x ∈ U. Then, giving A ∈ LR(H), we have 1) (QH )∗ = J H ⊥ , A(0) A(0)

2) (QA A)∗ = A∗ JA(0)⊥ , 3) D((QA A)∗ ) = D(A∗ ), 4) R((QA A)∗ ) = R(A∗ ). Lemma 2.2 Let A ∈ LR(H) and B ∈ LR(K). Let {yn } be an orthonormal sequence of N (B)⊥ such that kQB Byn k → 0. Suppose that there exists C ∈ LR(K, H) such that C(0) ⊆ A(0) for which R(MC ) is closed. Then there exist ǫ0 > 0 and a subsequence {ynk } of {yn } such that d(R(QA A), QA Cynk ) ≥ ǫ0 for all k ∈ N. Proof Since R(MC ) is closed, we have    

u  

      QMC MC u v   , γ(MC ) := inf ∈ / N (MC ) 6= 0.  u  v    dist  , N (MC ) v

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Suppose that for all ǫ > 0 and {ynk } subsequence of {yn } , dist(R(QA A), QA Cynk ) ≤ ǫ. Then dist(R(QA A), QA Cyn ) → 0. Hence there exists {xn }n∈N ∈ H such that kQA Axn − QA Cyn k → 0. For each n ∈ N, we can    xn prove that dist , N (MC ) ≥ 1. To see this observe that −yn           

xn xn u

, u ∈ N (MC ) dist , N (MC ) = inf −

−y −yn v v n     p u kxn − uk2 + kyn + vk2 , ∈ N (MC ) = inf v     p u = inf kxn − uk2 + kyn k2 + kvk2 , ∈ N (MC ) v ≥ kyn k = 1. Then

 

xn

QMC MC

 

xn −yn

  ≤  Q M M C C

xn −yn , N (MC ) dist −yn p = kQA Axn − QA Cyn k2 + kQB Byn k2 → 0, which is a contradiction with the fact that γ(MC ) 6= 0. 2 Remark 2.1 Let A ∈ LR(H). Then H/A(0) is a Hilbert space when equipped which the scalar product h, i1 defined by hx, yi1 := hP (x), P (y)i for all x, y ∈ H when P is the orthogonal projection with direction A(0).

3

Essential Approximate Point Spectra for Matrix of Linear Relations

We start our study with the following proposition which represents an improvement of the main theorem given by Xiaohong Cao [4] to the class of linear relations. Proposition 3.1 Let A ∈ BRφ+ (H) and B ∈ CBR(K) such that R(B) is not closed. Then d(A) = +∞ if and only if there exists C ∈ L(K, H) such that MC ∈ Rφ− + (H ⊕ K). ⊥ Proof Since d(A) = +∞, we have R(A) is an infinite separable Hilbert space. Then there exists an isometric T : K → R(A)⊥ . Define an operator C : K → H by     R(A)⊥ T . C = :K→ 0 R(A) Then we claim that MC is an upper semi-Fredholm relation with negative index.   u First, we will prove that MC is an upper semi-Fredholm relation. Let ∈ N (MC ), v then 0 ∈ Au + Cv and 0 ∈ Bv. Thus −Cv ∈ Au and hence −Cv ∈ R(A) ∩ R(A)⊥ = {0}.

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Since Cv = T v, then v = 0 and u ∈ N (A). This induces that N (MC ) ⊆ N (A) ⊕ {0}. Then n(MC ) ≤ n(A) < +∞. Now to prove that R(MC ) is closed, it suffices to prove that R(QMC MC ) is closed (see Cross [1, II.3.6 (pg. 35)]). Using Proposition 2.1, we have   QA A QA C . QMC MC =  0 QB B 

   xn u˜ → . Thus QA Axn + QA Cyn → u ˜ and yn v˜ QB Byn → v˜. Let un ∈ Axn . Then QA Axn = QA un . Since

Let (xn , yn ) ∈ H ⊕ K such that QMC MC

hQA un , QA Cyn i1 = hPA(0)⊥ (un ), PA(0)⊥ (Cyn )i = hun , Cyn i − hPA(0) (un ), PA(0) (Cyn )i = −hPA(0) (un ), PA(0) (Cyn )i = −hPA(0) (un ), Cyn i = 0, it follows that QA Axn ⊥ QA Cyn . Therefore both of {QA Axn }n∈N and {QA Cyn }n∈N are Cauchy sequences. By the definition of C, we have Cyn = T yn , and so p kT yn k2 + kuk2 = kT ynk = kyn k. kQA Cyn k = kQA T yn k = d(T yn , A(0)) = inf u∈A(0)

Hence {yn }n∈N is a Cauchy sequence. Let yn → y˜ andQAAxn → QA A˜ x. Then ˜ = QA A˜ x + QA C y˜ and v˜ = QB B y˜.  u x ˜ u ˜ , and hence R(MC ) is closed. Then the preceding = QMC MC It means that y˜ v˜ proof tells us that MC is an upper semi-Fredholm relation. Second, we will prove that i(MC ) ≤ 0. We know that R(MC ) ⊆ H ⊕ R(B). Therefore d(MC ) ≥ d(B) = +∞ (because R(B) is not closed) so that i(MC ) = n(MC ) − d(MC ) ≤ 0 (since n(MC ) < +∞). For the converse, assume that d(A) = N < +∞. Since R(B) is not closed, there exists an orthonormal sequence {yn } in N (B)⊥ such that kQB Byn k → 0. In fact, since R(B) is not closed, it follows that   kQB Byk γ(B) = inf , y∈ / N (B) = 0. d(y, N (B)) Let y ∈ K\N (B). Write y = y1 + y2 , where y1 ∈ N (B) and y2 ∈ N (B)⊥ . Then kQB Byk kQB Byk kQB By2 k = = d(y, N (B)) inf {ky1 + y2 − zk, z ∈ N (B)} inf {ky2 − zk, z ∈ N (B)} kQB By2 k kQB By2 k y2 np o = = = kB k. ky2 k ky2 k inf ky2 k2 + kzk2 , z ∈ N (B)

 Thus inf kQB Byk : kyk = 1 and y ∈ N (B)⊥ = 0.

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Hence there exists an orthonormal sequence (yn )n∈N ⊆ N (B)⊥ such that kQB Byn k → 0. Since R(MC ) is closed (because it is an upper semi-Fredholm relation), then, using Lemma 2.2, there exist ǫ0 > 0 and a subsequence {ynk } of {yn } for which d(R(QA A), QA Cynk ) ≥ ǫ0 for all k ∈ N. Then we can suppose, considering a subsequence, that d(R(QA A), QA Cyn ) ≥ ǫ0

for all n ∈ N.

(3.1)

Hence, by [4, (2.4) (pg. 765)], there exists a sequence (θn )n∈N ∈ [0, 2π] such that

X 

1 iθn

C yn e

= +∞,

n n≥0

P

but the fact

n≥1

2 P 1 iθn y e

= n n

n≥1

1 n2

< +∞, implies that y =

P

n≥1

1 n

yn eiθn ∈ K which is a

contradiction. Therefore we must have d(A) = +∞. 2 Proposition 3.2 Let A ∈ LR(H), B ∈ LR(K) and C ∈ LR(K, H) such that R(A) is closed and MC ∈ Rφ+ (H ⊕ K). If n(B) = +∞, then d(A) = +∞. Proof To the contrary, we suppose that d(A) < +∞. There are two cases to consider. Case 1 Suppose that C(N (B)) is finite dimensional. Then dim(N +∞,  ∩ N (B))  = (C) 0 0 for ∈ MC thus there exists an orthonormal sequence {yn } ∈ N (C) ∩ N (B). Hence yn 0 each n, which implies that N (MC ) is infinite dimensional. This is a contradiction. Case 2 Suppose that C(N (B)) is infinite dimensional. Since R(A) is closed, we know, by our assumption, that dim R(A)⊥ = d(A) < +∞. If dim(C(N (B)) ∩ R(A)) < +∞, then dim C(N (B)) < +∞, which is a contradiction. Therefore C(N (B)) ∩ R(A) is infinite dimensional. Thus we can find an orthonormal sequence {yn } ∈ N(B)  exists a  for which  there xn 0 for each ∈ MC sequence {xn } ∈ H such that Cyn ∈ Axn for each n. Then −yn 0 n, which implies that N (MC ) is infinite dimensional. It is a contradiction again. Therefore d(A) = +∞. 2 Proposition 3.3 Let A ∈ BRφ+ (H) and B ∈ CBR(K) such that R(B) is closed. If n(B) = d(A) = +∞, then there exists C ∈ L(K, H) such that MC ∈ Rφ− + (H ⊕ K). Proof Let N = n(A). There are two cases to consider: Case 1 Suppose d(B) = +∞. Since N (B) and R(A)⊥ are infinite dimensional separable Hilbert spaces, then there exists an isometric T : N (B) → R(A)⊥ . Define an operator C : K → H by:       C=

T

0

0

0

:

N (B)

N (B)⊥

→

R(A)⊥ R(A)

.

  u ∈ N (MC ), then 0 ∈ Au + Cv and v 0 ∈ Bv. Thus −Cv ∈ Au and so Cv ∈ R(A)∩R(A)⊥ . Then Cv = 0 and 0 ∈ Au. Since v ∈ N (B), we get that Cv = T v = 0. Therefore v = 0. Now we have proved N (M  that   C ) ⊆ N (A) ⊕ {0} un u ˜ and so n(MC ) ≤ n(A) < +∞. Now suppose that QMC MC → . Thus QA Aun + vn v˜ Then we claim that MC ∈ Rφ− + (H ⊕ K). In fact, if

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QA Cvn → u ˜ and QB Bvn → v˜. Write vn = αn + βn , where αn ∈ N (B) and βn ∈ N (B)⊥ . We claim that {βn } is a Cauchy sequence. Indeed, since QB Bvn = QB Bβn → v˜ and R(QB B) is closed, it follows that there exists v ∈ N (B)⊥ such that QB Bvn = QB Bβn → QB Bv = v˜ and hence QB B(βn − v) → 0. Since (βn − v) ∈ N (B)⊥ and γ(QB B) > 0, we get that βn − v → 0. Then {βn } is a Cauchy sequence. Also since {QA Aun + QA Cvn } is a Cauchy sequence and QA Aun ⊥ QA Cvn we know that {QA Aun } and {QA Cvn } are all Cauchy sequences. By the definition of C, Cvn = C(αn + βn ) = T αn and hence {αn } is a Cauchy sequence. Thus {vn } is a Cauchy sequence. Let vn → y˜ and x, then u ˜ = QA A˜ x + QA C y˜ and v˜ = QB B y˜.    QA Aun → QA A˜ u ˜ x ˜ It means that = QMC MC . Hence R(MC ) is closed by [1, II.3.6 (pg. 35)]. v˜ y˜ Since R(MC ) ⊆ H ⊕ R(B), it follows that d(MC ) ≥ d(B) and hence d(MC ) = +∞. Then MC ∈ Rφ− + (H ⊕ K). Case 2 Suppose d(B) < +∞. Then d(QB B) < +∞. Hence dim R((QB B)∗ ) = +∞. Indeed, if we suppose that dim R((QB B)∗ ) < +∞, then (QB B)∗ is a compact operator. By Schauder Theorem (QB B)∗∗ is also a compact operator. As R((QB B)∗∗ ) is a closed subspace of K/B(0) (because R((QB B)∗∗ ) = N ((QB B)∗ )⊥ ), then dim R((QB B)∗∗ ) < +∞. It follows that dim N ((QB B)∗ )⊥ < +∞. Thus dim N ((QB B)∗ ) = d(QB B) = +∞, a contradiction. From Lemma 2.1, dim R(B ∗ ) = +∞. Then dim N (B)⊥ = dim R(B ∗ ) = +∞. Let {e1 , e2 , · · · , eN } be an orthonormal set in N (B)⊥ and let M = span{e1 , e2 , · · · , eN }. Since (N (B) + M ) and R(A)⊥ are infinite dimensional separable Hilbert spaces, then there exists an isometric J : N (B) + M → R(A)⊥ . Define an operator C : K → H by       J 0 N (B) + M R(A)⊥ : → . C= 0 0 R(A) (N (B) + M )⊥ − Then we claim  that MC ∈ Rφ+ (H ⊕ K). u ∈ N (MC ), then 0 ∈ Au + Cv and 0 ∈ Bv. Thus −Cv ∈ Au and so Indeed, if v Cv ∈ R(A) ∩ R(A)⊥ . Hence Cv = 0 and 0 ∈ Au. Since v ∈ N (B) + M, we get that Cv = Jv = 0. Therefore v = 0 because J is injective. Thisproves  N (MC ) = N (A) ⊕ {0}, so  that u ˜ un . Thus QA Aun + QA Cvn → u ˜ → n(MC ) = n(A) = N. Now suppose that QMC MC v˜ vn and QB Bvn → v˜. Thus {QA Aun } and {QA Cvn } are all Cauchy sequences. Write vn = αn + βn , where αn ∈ N (B) + M and βn ∈ (N (B) + M )⊥ . By the definition of C, Cvn = J(αn ), and hence {αn } is a Cauchy sequence. We claim that {βn } is a Cauchy sequence. Indeed, since QB Bβn = QB Bvn − QB Bαn , it follows that {QB Bβn } is a Cauchy sequence. Since (N (B) + M )⊥ ⊆ N (B)⊥ and R(QB B) is closed, we can get that {βn } is a Cauchy sequence. Therefore {vn } isa Cauchy sequence. Let vn → y˜ and QA Aun → QA A˜ x, it means that    u ˜ x ˜ = QMC MC and hence R(MC ) is closed, by [1, II.3.6 (pg. 35)]. v˜ y˜ We can that d(MC ) = N. To see this, we need to prove that n(MC∗ ) = N.   also prove  xn Let : 1 ≤ n ≤ k ⊆ N (MC∗ ) be an orthonormal set. Then A∗ xn = 0 and hence xn ∈ yn N (A∗ ) = R(A)⊥ . By definition of C,

C ∗ xn = J ∗ xn = −B ∗ yn ∈ (N (B) + M ) ∩ N (B)⊥ = M.

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Observe that J ∗ is injective, then k ≤ N and hence n(MC∗ ) ≤ N. Since en ∈ N (B)⊥ ∩ M for each n ∈ {1, 2, · · · , N }, then there exist un ∈ R(A)⊥ = N (A∗ ) ∗ ∗ and vn ∈R(B ∗ ) such   that en = J un = B vn . Then {u1 , u2 , · · · , uN } is linear independent and un , n ≤ N ⊂ N (MC∗ ). It implies that n(MC∗ ) ≥ N. Therefore n(MC∗ ) = N = d(MC ). −vn It means that MC is Weyl. Then MC ∈ Rφ− 2 + (H ⊕ K). Proposition 3.4 Let A ∈ BRφ+ (H) and B ∈ BRφ+ (K). Then n(A) + n(B) ≤ d(A) + d(B) if and only if MC ∈ Rφ− + (H ⊕ K) for some C ∈ L(K, H). Proof Suppose n(A)+n(B) ≤ d(A)+d(B). We can prove that for all C ∈ LR(K, H) such that C(0) ⊆ A(0), MC ∈ Rφ+ (H ⊕ K). To see this, let C ∈ LR(K, H) verifying C(0) ⊆ A(0). Write    I 0 A C MC = . 0 B 0 I     I 0 A C Denote by D1 = and D2 = . 0 B 0 I   x Let ∈ N (D1 ), then x = 0 and 0 ∈ By. Hence N (D1 ) ⊆ {0} ⊕ N (B) so n(D1 ) ≤ y n(B) < +∞. Nowto provethat R(D1 ) is closed, it suffices to prove that R(D1∗ ) is closed. I 0 , it follows that d(D1∗ ) = d(B ∗ ) = n(B) < +∞. Then R(D1 ) is Since D1∗ = 0 B∗ closed. Similarly, we can prove that D2 ∈ Rφ+ (H ⊕ K). Which implies that MC is an upper semi-Fredholm relation. On the other a single valued relation, it follows that i(MC ) ≤ 0. In C is hand, since A C I 0 then using the index theorem for the product of linear deed i(MC ) = i 0 I 0 B relations seen by Cross [2],     −1   A C 0 I 0 0 +i  − dim   ∩      B 0 I 0 I 0 0B 0      0 IC A0 +i +i  B 0 I 0 I



I 0



I

i(MC ) = i 

= i

0

0





AC



= i(B) + i(A) ≤ 0.

2 As a consequence of Propositions 3.1, 3.2, 3.3 and 3.4, we obtain this fundamental result. Theorem 3.1 Let A ∈ CBR(H) and B ∈ CBR(K). MC ∈ Rφ− + (H ⊕ K) for some C ∈ L(K, H), if and only if A ∈ Rφ+ (H) and     • n(B) < +∞ and n(A) + n(B) ≤ d(A) + d(B)   

or n(B) = d(A) = +∞,

• d(A) = +∞,

if R(B) is closed,

(3.2)

if R(B) is not closed.

Corollary 3.1 Let A ∈ CBR(H) and B ∈ CBR(K). MC ∈ Rφ+ − (H ⊕ K) for some

No.4

S. Elleuch & M. Mnif: ESSENTIAL APPROXIMATE POINT SPECTRA

C ∈ L(K, H) if and only if B ∈ Rφ− (K) and     • d(A) < +∞ and d(A) + d(B) ≤ n(A) + n(B)   

or d(A) = n(B) = +∞

• n(B) = +∞

if R(A) is closed ,

1197

(3.3)

if R(A) is not closed .

Proof This result is obtained by interchanging the roles of MC and MC∗ . 2 Combining Theorem 3.1 and Corollary 3.1 we obtain Corollary 3.2 Let A ∈ CBR(H) and B ∈ CBR(K). MC is Weyl for some C ∈ L(K, H) if and only if A ∈ Rφ+ (H), B ∈ Rφ− (K) and   d(A) < +∞, n(B) < +∞ and n(A) + n(B) = d(A) + d(B) (3.4)  or n(B) = d(A) = +∞.

As a consequence, we give the main result of this section. Theorem 3.2 Let A ∈ CBR(H) and B ∈ CBR(K). If σea (A) = σSF + (B) or σSF − (A) ∩ σSF + (B) = ∅, then for all C ∈ L(K, H), σea (MC ) = σea (M0 ). Proof Let C ∈ L(K, H) and λ ∈ C. Write     I 0 I C A − λI 0   . MC − λI =  0 B − λI 0 I 0 I       A − λI 0 I C I 0 . and D3 = , D2 = We denote D1 = 0 I 0 I 0 B − λI We have   A − λI 0  = D1 D3 . M0 − λI =  0 B − λI

For the first inclusion, let λ0 ∈ ρea (M0 ). Then M0 −λ0 I ∈ Rφ− + (H⊕K). Thus D3 ∈ Rφ+ (H⊕K) see (Cross [1, V.2.16]).   x ∈ N (D1 ). Then x = 0 We can also prove that D1 ∈ Rφ+ (H ⊕ K). Indeed let y     x 0 . Hence N (D1 ) ⊆ N (M0 ). Thus ∈ (M0 − λ0 I) and 0 ∈ (B − λ0 I)y. Therefore y 0 n(D1 ) ≤ n(M0 ) < +∞. Since R((M0 − λ0 I)∗ ) ⊆ H ⊕ R((B − λ0 I)∗ ) = R(D1∗ ), then d(D1∗ ) ≤ d((M0 − λ0 I)∗ ) = n(M0 − λ0 I) < +∞. Which implies that D1 ∈ Rφ+ (H ⊕ K). As D2 is invertible, then MC − λ0 I ∈ Rφ+ (H ⊕ K) with i(MC − λ0 I) = i(M0 − λ0 I) ≤ 0. Now we will prove that σea (M0 ) ⊆ σea (MC ). Suppose that σea (A) = σSF + (B). If MC − λ0 I ∈ Rφ− + (H ⊕ K), then A − λ0 I ∈ Rφ+ (H). Using Theorem 3.1, we know that if λ0 ∈ σSF + (B), then d(A − λ0 I) = +∞ and hence A − λ0 I ∈ Rφ− / + (H). It follows that λ0 ∈ σea (A), which is a contradiction. Then B − λ0 I ∈ Rφ+ (K). Therefore M0 − λ0 I ∈ Rφ+ (H ⊕ K) with i(M0 − λ0 I) = i(MC − λ0 I) ≤ 0 which means that λ0 ∈ / σea (M0 ). Similarly, if σSF − (A) ∩ σSF + (B) = ∅, the result is true also. 2

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Vol.33 Ser.B

Fredholm n × n Upper Triangular Matrix of Linear Relations

In this section we will generalize the results obtained on the last section, in the case of n×n upper triangular matrix of linear relations with n ≥ 2. Let H1 , H2 , · · · , Hn denote n infinite dimensional separable Hilbert spaces with n ≥ 2. Let Ai ∈ LR(Hi ) for 1 ≤ i ≤ n and Cij ∈ LR(Hj , Hi ) for 1 ≤ i < j ≤ n. We denote by L Mn the linear matrix relation acting on Hi , of the form 1≤i≤n

Mn = ((Mn )ij )1≤i,j≤n     Cij , if i < j; with (Mn )ij = Ai , if i = j;    0, if i > j. With the same manner then in the case of 2 × 2 matrix of linear relations, we can prove Proposition 4.1 Let Ai ∈ LR(Hi ) for all 1 ≤ i ≤ n and Cij ∈ LR(Hj , Hi ) such that L Cij (0) ⊆ Ai (0) for all 1 ≤ i < j ≤ n. Then QMn Mn is the linear operator acting on Hi , 1≤i≤n

of the form

QMn Mn = ((QMn Mn )ij )1≤i,j≤n

with (QMn Mn )ij =

    QAi Cij ,   

QAi Ai ,

if i < j; if i = j;

0, if i > j. Theorem 4.1 Let A1 ∈ BRφ+ (H1 ) and Ai ∈ CBR(Hi ) for all 2 ≤ i ≤ n such that R(An ) is not closed. Then d(A1 ) = +∞ if and only if there exists (Cij )1≤i
Proof Since d(A1 ) = +∞ then dim R(A1 )⊥ = +∞. Hence there exists an isometric T : H2 × H3 × · · · × Hn → R(A1 )⊥ .

We define for all 2 ≤ j ≤ n; 

C1j : Hj −→ 

R(A1 )⊥



x −→ C1j x =  verifying for all x = (x2 , · · · , xn ) ∈

Q

2≤i≤n

Hi ,

R(A1 )

 

T (0, · · · , x, · · · , 0) 0 P



;

C1j (xj ) = T (x).

2≤j≤n

Then we claim that for C1j defined in this way and for all Cij ∈ LR(Hj , Hi ), 2 ≤ i < j ≤ n, Mn is an upper semi-Fredholm relation. We first show that n(Mn ) < +∞. P Let x = (x1 , · · · , xn ) ∈ N (Mn ). Then 0 ∈ A1 x1 + C1j (xj ). 2≤j≤n

No.4

S. Elleuch & M. Mnif: ESSENTIAL APPROXIMATE POINT SPECTRA

Thus −

P

1199

C1j (xj ) ∈ A1 x1 . Hence −T (x2 , · · · , xn ) ∈ A1 x1 . So T (x2 , · · · , xn ) ∈ R(A1 ) ∩

2≤j≤n

R(A1 )⊥ = {0} . Then T (x2 , · · · , xn ) = 0 and 0 ∈ A1 x1 . This proves that N (Mn ) ⊆ N (A1 ) ⊕ {0} ⊕ · · · ⊕ {0} . Then n(Mn ) ≤ n(A1 ) < +∞. e where X e = (f Now let Xm = (x1m , x2m , · · · , xnm ). Suppose that QMn Mn Xm → X x1 , f x2 , P P ···,x fn ). Thus QA1 A1 x1 m + QA1 C1j xj m → f x1 . Since QA1 A1 x1 m ⊥ QA1 C1j xj m , 2≤j≤n 2≤j≤n P then both of {QA1 A1 x1 m }m∈N and { QA1 C1j xj m }m∈N are Cauchy sequences. P2≤j≤n By the definition of C1j , we have C1j xj m = T (x2 m , x3 m , · · · , xn m ), then 2≤j≤n

 X 

QA1 C1j xj m

= kQA1 T (x2 m , x3 m , · · · , xn m )k 2≤j≤n

= dist(T (x2 m , x3 m , · · · , xn m ), A1 (0))

= =

inf

u∈A1 (0)

inf

u∈A1 (0)

kT (x2 m , x3 m , · · · , xn m ) − uk p kT (x2 m , x3 m , · · · , xn m )k2 + kuk2

= kT (x2 m , x3 m , · · · , xn m )k = k(x2 m , x3 m , · · · , xn m )k.

Which implies that for all 2 ≤ j ≤ n, (xj m )m∈N is a Cauchy sequence. Let QA1 A1 x1 m → QA1 A1 a1 and xj m → aj for all 2 ≤ j ≤ n. 

a1



       a2  P  e Then X = (QAi Ai ai + QAi Cij (aj )), QAn An an = QMn Mn   ..  . i 0 and a subsequence {tmk }k∈N of {tm }m∈N for which d(R(QA1 A1 ), QA1 C1n tmk ) ≥ ǫ0

for all k ∈ N.

Assume to the contrary that d(R(QA1 A1 ), QA1 C1n tm ) → 0 as m → +∞. Thus there exists {xm } ∈ H1 such that kQA1 A1 xm − QA1 C1n tm k → 0.

1200

ACTA MATHEMATICA SCIENTIA

L

We denote by Xm = (xm , 0, · · · , 0, −tm ) ∈

Vol.33 Ser.B

Hi . It is easy to see that dist(Xm , N (Mn ))

1≤i≤n

≥ 1. Then

kQMn Mn Xm ≤ kQMn Mn Xm k dist(Xm , N (Mn ) p = kQA1 A1 xn − QA1 C1n tm k2 + kQAn An tm k2 → 0. Which is in contradiction with the fact that γ(Mn ) 6= 0. So, there exist ǫ0 > 0 and a subsequence {tmk }k∈N of {tm }m∈N such that d(R(QA1 A1 ), QA1 C1n tmk ) ≥ ǫ0 for all n ∈ N. And hence there exists a sequence (θm )m∈N ∈ [0, 2π] such that

 X

1 iθm

QA1 C1n t e m

= +∞.

m m≥1

P

But the fact

m≥1

2

1 iθm

m tm e

=

P

m≥1

1 m2

< +∞, implies that t =

P

m≥1

1 iθm m tm e

∈ Hn .

A contradiction. Therefore we must have d(A1 ) = +∞. 2 Theorem 4.2 Let A1 ∈ BRφ+ (H1 ) and for all 2 ≤ i ≤ n, Ai ∈ CBR(Hi ) such that R(Ai ) is closed. If d(Ai−1 ) = +∞, for all i ∈ {2, 3, · · · , n} such that n(Ai ) = +∞, then for all 1 ≤ i < j ≤ n, there exists Cij ∈ BR(Hj , Hi ) such that Mn ∈ Rφ+ (

L

Hi ).

1≤i≤n

Proof We proceed by induction. The case n = 2 is proved in the last section. Suppose that this result holds in the order n and for all i ∈ {2, · · · , n + 1} , R(Ai ) is closed. We need to prove that if: d(Ai−1 ) = +∞, for all i ∈ {2, · · · , n + 1} such that n(Ai ) = +∞, then there exists Cij ∈ BR(Hj , Hi ) such that Mn+1 ∈ Rφ+ (

L

Hi ).

1≤i≤n+1

Construction of (Cij )1≤i
M

Hi ).

1≤i≤n

Construction of (Ci(n+1) )1≤i≤n L We have Mn ∈ BRφ+ ( Hi ) and R(An+1 ) is closed. Therefore: 1≤i≤n

If n(An+1 ) < +∞, then for all C = (Ci(n+1) )1≤i≤n , 

Mn+1 = 

Mn

C

0

An+1



 ∈ Rφ+ (

M

1≤i≤n

Hi ).

No.4

S. Elleuch & M. Mnif: ESSENTIAL APPROXIMATE POINT SPECTRA

1201

If n(An+1 ) = +∞, then from the inductive assumption, d(An ) = +∞. As d(An ) ≤ d(Mn ) then d(Mn ) = +∞ which implies from Proposition 3.3, that there exists C = (Ci(n+1) )1≤i≤n L such that Mn+1 ∈ Rφ+ ( Hi ). 2 1≤i≤n+1

Theorem 4.3 Let A1 ∈ BRφ+ (H1 ) and for all 2 ≤ i ≤ n, Ai ∈ CBR(Hi ) such that R(An ) is closed. Suppose that there exists i ∈ {2, 3, · · · , n − 1} such that R(Ai ) is not closed. Let p = inf {i ∈ {2, 3, · · · , n − 1} such that R(Ai ) is not closed } . If

d(A1 ) = +∞ and d(Ai−1 ) = +∞, for all i ∈ {p + 1, · · · , n} such that n(Ai ) = +∞, L then for all 1 ≤ i < j ≤ n, there exits Cij ∈ BR(Hj , Hi ) such that Mn ∈ Rφ+ ( Hi ). 1≤i≤n

Proof With the same manner then in the proof of Theorem 4.2, we suppose that this result holds in the order n. Suppose that R(An+1 ) is closed and that there exists i ∈ {2, 3, · · · , n} such that R(Ai ) is not closed. Let p˜ = inf {i ∈ {2, 3, · · · , n} such that R(Ai ) is not closed } . Construction of Cij , 1 ≤ i < j ≤ n If p˜ ≤ n − 1. In this case, there are two cases to consider: Case 1 If R(An ) is closed then the fact of d(A1 ) = +∞ and d(Ai−1 ) = +∞, for all i ∈ {˜ p + 1, · · · , n} such that n(Ai ) = +∞, implies, from the inductive assumption, that there exist Cij , 1 ≤ i < j ≤ n such that Mn ∈ L Rφ+ ( Hi ). 1≤i≤n

Case 2 If R(An ) is not closed, then since d(A1 ) = +∞, it follows, from Theorem 3.1, L that there exist Cij , 1 ≤ i < j ≤ n such that Mn ∈ Rφ+ ( Hi ). 1≤i≤n

Now if p˜ = n, it reduces to the Case 2. Gathering this two cases into one. We obtain

if d(A1 ) = +∞ and d(Ai−1 ) = +∞ for all i ∈ {˜ p + 1, · · · , n} such that n(Ai ) = +∞, L then there exits Cij , 1 ≤ i < j ≤ n such that Mn ∈ Rφ+ ( Hi ). 1≤i≤n

Construction of C = Ci(n+1) , 1 ≤ i ≤ n L Since Mn ∈ BRφ+ ( Hi ) and R(An+1 ) is closed. Then 1≤i≤n

if n(An+1 ) < +∞ or n(An+1 ) = d(An ) = +∞, then, by Proposition 3.3, there exists C = Ci(n+1) , 1 ≤ i ≤ n such that M Mn+1 ∈ Rφ+ ( Hi ).

2

1≤i≤n+1

References [1] Cross R W. Multivalued Linear Operators. New York: Dekker, 1998 [2] Cross R W. An index theorem for the product of linear relations. Linear Alg Appl, 1998, 277: 127–134 [3] Azizov T Y, Behrndt J, Jonas P, Trunk C. Compact and finite rank perturbations of linear relations in Hilbert spaces. Integral Equations Operator Theory, 2009, 63: 151–163 [4] Cao X H, Meng B. Essential approximate point spectra and Weyl’s theorem for operator matrices. J Math Anal Appl, 2005, 304: 759–771