Arándiga, F.

Caselles, V.

Portal de Revistas Científicas Complutenses

REVISTA MATEMÁTICA de la
Universidad Complutense de Madrid
Volumen 7, número 2: 1994
Qn Strongíy Stabíe Approxirnations
F. ARÁNDIGA aud V. CASELLES’
AHSTRÁCT. In this paper we prove that the convergence of (7’ —
to zero in operator norm (Plus sorne technical conditions) is a sufflcient condi-
tion for T,, to be a strongly stahle approximation to 7’, thus extending sorne
previous resulte existing jo tite literature.
1. INTRODUCTION
Let 7’ be a bounded linear operator on a complex Banach space
X and A be an isolated nonzero eigenvahie of 7’ with finite algebraic
ínultiplicity. Let 7’,, be a sequence of bounded linear operators on X.
In titis paper we give sufflcient conditions under whicb 7’,, is a strongly
stable approximation of 7’ at A. This work is motivated by recent results
of M. Ahues [1], R. Bouldin [4] aud M. Titambar Nair [10]. Our effort is
directed to establish a minimal set of assuniptions to guarantee tlxat T,,
is a strongly stable approximation of 7’ and A in tite spirit of tite aboye
níentioned works.
1
Partially supported by D.G.I.C.Y.T. project PS9O-0271.
1991 MatLie,natics Subject Cíassification: 65F15, iSAiS
Editorial Conxplutense. Madrid, 1994.
http://dx.doi.org/10.5209/rev_REMA.1994.v7.n2.17729
208 F. Arándiga and V. Caselles
2. PRELIMINARIES AND STATEMENT OF THE MAIN
RESULT
In the sequel let X be a complex Banach space and let £(X) be
tite space of bounded linear operators on X. The spectrum of 7’, i.e.
tbe set of z E C such that zí — 7’ is not invertible in £(X) will be
denoted by u(T). If z E p(7’) C a(T), tite resolvent set, R(z, 7’)
(U — T)’ E £(X).
A E a(T) is a Riesz point of a(T) if A is a pole of the resolvent
R(z,T) with a residnuin P = ~ f~. R(z,T)dz of Imite rank (risa curve
on tite complez plane around A containing A as the only singularity of
R(z,T)). Tite Riesz points ofa(T) are tite isolated eigenvalues of 7’ with
finite algebraic ínultiplicity. IfA E a(T) is not a Riesz point of «(7’), we
write A E
Let us explain an nltrapower construction introduced by .9. Berbe-
rían witich is used in spectral theory ([9] VA) Let us Ilx an ultrafilter U
on N containin&the Frecitet filter. The ultrapower of X witit respect to
U, denoted by X¿.~ or simply by X is defined by V~(X)/Cu(X) witere
t”(X) := {(crn)u : x,, E X, sup IIxnII < +~}
n
ancl
{(x,1) c t~(X) hm IIx,,II = OFu
Operators on X can be Iifted to operators on 2 by T~ = (Tx,~)u,
£ E 2, 7’ E £(X) in such a way that a(1) = «(7’), ([9], Theorern
V.1.4). The approximate point spectrnm of 7’ is converted into tite
point spectrum of T ([91, Theorem. V.1.4). Now, we recahl tite following
result froin [6].
Theorem 2.1. Let 7’ E .C(X). La 8,,a(T) tite exterior bozíndary
of«(7’) (= tite boundary of tite unbounded connected component ofp(T)).
Titen ¿9~a(T) o a~,,(7’) = &~a(T) O {z E C : dim Ker(z — 7’) is
ittfiitite }.
Titis result means that tite elgenspace associated to a Riesz point on
tite exterior boundary of «(7’) is contained in X and cannot be enlarged
by going to X.
On Strongly Stable Approximations 209
Tite following definitions are taken from [7].
Definitian 2.1. Let 7’, 7’,, E £(X) n = 1,2 Tite stabie con-
vergence of 7’,, to 7’, denoted by 7’,, -% 7’, means that:
1. T»x —* 7’x in X for aH z E X
2. ~M >0, BN EN such that Vn> N 7’—1 E £(X) andIIT;’II =
M.
Definition 2.2. LetT,T,, E 12(X) u = 1,2 Let A be a Riesz
point of «(7’) of algebraic multipiiciiy ni. Wc soy that 7’,, is strongly
stable convergent to 7’ at A and denote it by 7’,, —~t 7’ at A if titere
exists a ciosed Jordan curve 1’ (2 p(2’) around A such that A is tite oniy
spectral point of 7’ inside r and sucit that
1. 7’,, — 2 .~t4 7’— z, for di z E r
2. dirn P,,X = ni br u large enouglz, where 1% is tite spectra¿
projection associated mnith «(7’,,) o A (A denotes tite set ofpoints z E C
inside F).
Here one sitould mention tbat if 7’,, —z —~-~ 7’— z, Vz E 1’, titen br
soine N = N(I’) E N and u> N «(7’,,) PA is an spectral set of «(7’,,)
and we may define 1’,, = ~ f~. R(z,T,,)dz ([7] p. 231,232).
Tite strong stability of 7’,, at A isa sufficient condition for tite conver-
gence of tite eigenvalues of 7’,, inside 1’ with preservation of tite algebraic
multiplicities ([7] Titeorem 6.7).
Let us recal] the resnlts of M. Aitues [1], R. Bonldin [4] and M.
Titambar Nair [10]mentioned in tite introduction.
Let 7’,7’,, E 12(X) and let us suppose titat
(1) T,,x —* fEz for ah x E X.
(II) ¡¡(7’— T07’~¡I —* O as u —~
In [1], Ahites proved titat if 7’ E 12(X) is acompact operator sat-
isfying (1) and (II) and A ~ O is an elgenvaine of 7’, titen 7’,, 2 7’ at
A.
210 F. Arándiga and V. Caselles
Titen R. Bonidin [4]eliminated tite campacness of 7’ snpposing thaI
A / O is a Riesz point of «(7’) and adding tite assumption
(III) ¡ff47’ — T,,)¡¡ — O si u —*
Later, M. Titambar Nair [10] proved titat ib T and 7’,, E 12(X)
satisfy (1), (II), A # O is a Riesz point of «(7’) and one of tite following
assnmptions
(IV) 11(7’ - T4T11 —~ O
(V) IIfi’47’ - 7’~)II —~ O
(VI) ~N e N sncit titat Vn> N 7’,, is compact
(VII) BN E N siicb titat Vn > N 7’ — 7’,, is compact holds, titen
7’,, -t T at A.
Qur purpose is to prove tite following resnlt.
Theorem 2.2. Let T,7’,, E 12(X), u = 1,2,... be such that:
1. T,,x —~ fEz for aH z E X
2. ¡¡(7’— 7’,,)fZ’~FI —.0 zf u —* +~, for sorne k EN
and let A e &~«(fl, A ~ 0, be a Riesz point of «(7’). Then,
atA.
Titus, tite assumption titat A E Ooc,«(7’) (witicit we believe of no
harin in practice), permits 118 to econornize tite set of assumptions in tite
j)revious results.
An interesting consequence of Theorein 2.2 is:
Theorem 2.3. Let T,T,, E 12(X), u = 1,2,... be sueh that:
1. T,,x —* fEz for oil x E X
~ — 7’~)7’,tII —~ o as it —. ±~, for sorne k EN
and íd A E &,«(7’), A $ 0, be a Riesz point of «(7’) of algebraic
muitiphczty ni amI indez u _ ni. Titen, for sorne circie r of radius
r> O amundA, titere are exactty ni elgenvalues of 7’,, inside 1’ (counted
On Strongly Stable Approximations 211
according to their multiplicities) amI if «,, = {z E «(7’): jz — Aj < 4,
titen
max ¡z — Al” =C II7’~ — 7’IIX(T,X)
inhere X(T,A) is tite spectralsubspace ofX associated to A amI II.IIx(TA)
denotes tite operator non restnicted to X(7’, A).
Now, we go directly into tite proof of Theorem 2.2.
3. PROOF OF THE MAIN RESULT
First of ah we recall tite following result proved by M. Thamban
Nair [10]. We incinde its proof itere for tite sake of completeness.
Propositian 3.1. ¡Set T,7’,, E 12(X), u = 1,2,... aud let A # O
be a Riesz point of «(7’). Let 1’ be a closed Jordan curve aronud A
separating A frorn «(7’) -~ {A}. Suppose that:
1. 7’,,x —* 7’z fon aH z E X
2. ¡¡(T—7’QT4fl¡---.0asn—*+~,forsornekeN
3. T,,~z~L*T~z VzE 1’.
Titen 7’,, —~t 7’ atA.
Proof. Let
P= ¿JR(z,T)dz
and for sorne N = N(I’) and ahí u> N,
Fn = ¿jR(Z~Tn)dZ.
Tite result will follow if we prove tbat ¡ (E — P,,)P ¡j —. O and
(P — —-.0.
Since
R(z,T) — R(z,T,,) = R(z,T,,)(T — T,JR(z,T) =
212 F. Arkndiga and V. Caselles
= R(z,T)(T —
titen
(P-F,,)P=
(F — PQP,, = ~Jr
R(z,7’,,)(7’ — 7’,,)R(z,7’)Pdz
R(z,7’)(7’ — 7’,,)R(z,7’,,)F,,dz.
Since 7’,, — z ~ 7’— z Vz c 1’, then for some N = N(í’) E N
sup IlR(z~7’~)II <+00
zEr
n>N
(1)
([7], Cor. 5.2). It follows titat for some M > 0, I(~ — P,,)~II =
MII(T — 7’tFII. Now since 7’,,z —* Tz for all x E X and 1’ is a
projection of Imite rank, II(7’—7’,,)FII —* 0. llence II(F— I-’~)~II —. 0.
In tite saníe way, II(P — P,,)F,,¡¡ =Mfl(7’ — T,,)F,,¡¡ for sorne M > 0.
Now
(7’ Tn)Pn~$j (7’— T,,)R(z,T,,)dz
and nsing T,,R(z,T,.) = zR(z,T,,) —1,
(7’-T,,)P,,=
dz
(7’— 7’,,)(7’,,R(z, 7’,,) + 1)—.
2
Since A $ 0, we may suppose without loss of generality titat O is not
inside 1’. Titen fdz. O and
(T—7’,,)P,,=
dz
T,,)T,,R(z,T,,)—.
z
Proceeding indnctively,
f dz4;(7’— 7’,,)P,, = ~(7’— 7’n)7’,tR(2,7’,,)—r.
ir Z
On Strongly Stable Approxim.tions. 213
Using assumption (2) azul (1) it follows titat ¡¡(7’—fZX)P~I¡ — 0. Hence
— P,I)P,ZI¡ —*0. The proof follows in a standard way: it is easy to
see tbat tbe gap between X,, = P,.X and Xo = PX ([8],p. 197 for the
definition of gap), denoted by y(X,,, Xo) does not exceed
niax{II(P — J%)F¡¡, ¡¡(P —
From ¡¡(P» — ~)~II,IF(P~— P)P,,¡¡ —* Oit follows that -y(X,,,Xo) .c 1
for u large enough. According to ([8], p. 200 Cor. 2.6) this implies titat
duX,, = duX0 azul tite proposition is proved (see [41,p. 8).
Lemma 3.1. ¡Set T,7’,,, u = 1,2,... aud A be as in Titeorem 2.2.
¡Set A,, E O~«(7’,,) be such that A,, —* A. Titen, there exists u0 E N
suclz that A,, isa Rieszpoint of «(7’,,), Vn =u0.
Proof. Let E,, = Ker(A,, — 2’,,). Suppose titat dim E,, = +00 for
afl u E ¡ ¡ (2 N infinite. Without loss of generahity 1 = N. Let
~ tx~(F) = {Qa,,)u: £5,, EF,, Vti}.
Cu(F,,)
Then dim E = oo([5], Titeorem 3.1). Let £ E F, £5 = (½)u# 0 and
y,, E E,1. Hence 7’0~5» = A,,£5,,. Since IT,~~’ — Tflfl¡ —* 0 it foflows
that
= (A~~’ii,,)u = (TT$,,)u Ak7’ii.
Rut A / O, hence ff15 = A£5. TEtis E § Ker(A — T). Then
dim Ker(A — 7’) = +00. By Theorern 2.1, A cannot be a Riesz point of
«(7’). This contradiction proves tbat for sorne no E N and all u> u0
dirn Ker(A — T,,) <+00. Again by Theorem 2.1 Vn> u0, A,, is a Riesz
point of «(7’,,).
Lemma 3.2. ¡Set 7’,T,,, n= 1,2,... aud A be as Theorern 2.2.
Titen, Mere exist no E N, p > O sucit that for alt u =u0, «(7’,,) o
B(A, p*) consists of a firtite number of Riesz points of «(7’,,), whicit do
not accurnuiate on t9B(A,p~).
214 F. Arándiga and V. Caselles
Pronf. Since A E &,,u(T) is a Riesz point of «(7’), titere exists
a curve y : [O,+oo] —~ e such tital y(O) = A, y(+00) = 00 and
y(]O,+ocd) C p(7’). For eadx O > O let
{z E C : disi(z,y(]O,+ac[)) < 6}.
Notice that for sorne 6o > O, V60 o «(7’) = {A}. Write V = Vsa. Let
p> O and let ~/» := B(A,p) O y and 1/2k, be tite compleznent of 14,, in
1/. Titen V = 1/1,, u 1~2,,. Withont loss of generality, we suppose that
O « V. Tite proof proceeds in a series of claims.
Claim 1. Vp > O, p <
6o, Bn E N such that Vn =u% «(fE,,) O
= 0.
Otherwise, titere exists p > O, ,o < bo sncit tbat for eacit u E
N Bu’ > u and a point A,,> E «(7’,,’) O 1/2,,. Since fE,,> is uniformly
bonnded, {A,,~} is abounded sequence. Cali {A,,’} again as {A,,}. With-
atÉ loss of generality we suppose titat A,, —* ji E V2,,. Now it is easy
to check that ji E «(T). In fact we can take A,, E O~~«(7’~). Hence
= A,,£,, for sorne £,, E X, fl£,j¡ = 1. Titen
— ?k+í£,, = ?ft~£,, = A~T,ZZ..
Titus ff4,, = >.4,,. Hence f1£,,)u = i’-d£n)u. Titerefore jí E «ch =
«(7’), a contradiction since I~
2,, o «(7’) = 0.
Claim 2. Bp E]O,po[, Buo E N sucit titat Vn =no, «(7’,,) o B(A,p)
contains only a finite number of points.
Otberwise: Vp E]O,po[, Vn c N Bit’> u sncit titat a(7’4nB(A,p)
contains an infinite number of points. Using claim 1, we construct se-
quences pi,ni, snch tbat Pi+z < Pi, Pi — O, u1 < n~+í, u1 —* +00
sucb that
- «(1,,) 0 B(A,p1) contains an infinite number of points ami
- «(7’,,) 0 = O for a-II u> u1
Notice that titis implies titat the approximated point spectrum of
T,,~ in B(A,pfl (~ ~4«(7’,,~)O B(A,p~)) contains an infinite number of
On Strongly Stable Approximations 215
points. Let A,,• E Ae4T,,~) fl B(A,m) be an accurnula-tion point. Titen,
dim Ker(A,~~ —7’,,,) = 00 (Theorern 2.1 aboye). Let
F,¾=Ker(A,,~ — T,,~). As usuah we prove tbat
É= _____ C Ker(A-T).
Hence
dii Ker(A —7’) = 00,
a contradiction. TEtis, claim 2 is preved.
Claim 3. Bp > O(p < p), 2n5 E N, u5 =no such that Vn >
n~, «(7’,,) 0 B(A,p) contains only a finite number of Riesz points of
«(2’,.).
Otherwise: Vp > o(p* < p), Vn~ =u0 exists n’ =u~ sucit that
«(7’,,’) o B(A,p
5) conta-ins points witich are not Riesz points of «(7’,,>).
Inductively and using c]aims 1 amI 2 aboye we construct sequences
P~; -1 0, i¿~ 1+00, A,,~ E Oc’o«(Tn¿) O B(A,p~.) where A,,. is not a Riesz
point of «(7’,,~). Titis contradicts Lemma 3.1.
Fina-Uy, let A,, E «(fi’,.) O B(A,p), A,, —* ji. Without loss of
generality p~ is such that «(2’) o B(A,p) = {A}. Now, it is ea-sy to
prove that pi = A. Tite lemma is preved.
Remark. Te prove Titeorem 2.2 tite fuil strength of Lemma 3.2
aboye is not needed. In spite of it, we consider it interesting in itself to
be included here.
Fina-Uy let us sta-te wititout proof tite well known lemma:
Lemma 3.3. ([3] Proposition 2.1) ¡Set 7’ E 12(X) be .such titat foi’
sorne constant M > O
— T)xII =M¡¡xj¡ Vz E X, Vz E K
216 F. Arándiga and V. Caselles
inhere K is cornpact subset of C. ¡Set 7’,, E 12(X) be such that for sorne
k E N, 11(7’ — 7’~)7’,~II —~ O. Titen, BM1 > O and n0 E N snch that
¡j(z — 7’,,)z¡¡ =MzIIxIl Vz E X, Vz E 1<, Vn =no.
Praof of Theorem 2.2. Being 1’ = ¿9B(A,p
5), witit ? as ID
Leinma 3.2, let us prove tha-t T,,—z ~-* 7’—z, Vz E r. Then, Proposition
3.1 will do the rest. The only thing to be checiced is that for any z E
F, there exists M(z) > O, no(z) E N such that R(z,7’,,) exists and
¡¡R(z,7’,~)¡¡ =M(z), Vn =no(z). But titis is inmediate ID view of
Lemmas 3.2 azul 3.3 aboye.
References
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[2] Anselone, P.M.: Coflective¿y Compact Operator Approxirnatiou Tite-
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[3] Asadzaited, M.: ¡S~ and elgeuvaine error estirnates for discrete ordi-
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[4] Bonidin, R.: Operator Approxirnations with stable eigenvalues. J.
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[5] Boyd, M.W. and Transile, W.R.: Properties of ultraproducts. (Ren-
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397, 1979).
[6] Caselles, V.: Qn tite peripiteral spectrum of positive operators, Israel
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[7] Citatelin, F. :Spectral approxirnation of linear operators (Acadenuc
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[8] Kato, 1.: Perturbation tlzeory for linear operators (Springer-Verlag,
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On Strongly Stable Approximations 217
[9] Schaefer, ll.H.: Banach laltices amI positive operators (Springer-
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Departanxent de Matemática Aplicada
Astroixomia
Universitat de Valéncia
Dr. Moliner, 50
46100 Burjassot (Valencia)
SPAIN.
Departament de Matemátiques
informática
Universitat de les Illes Balears
Ctra. Valldemossa, Km 7,5
07071 Palma de Mallorca (Balears>
SPAIN.
Recibido: 17 de marzo de 1993
Revisado: 30 de septiembre de 1993


On Strongly Stable Approximations

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