On the spectral mapping theorem for essential spectra

Pub . Ma . UAB Vol . 29 Ns 2-3 No . 1985 ON THE SPECTRAL MAPPINGTHEOREM FOR ESSENTIAL SPECTRA M . Gonzalez and V . M . Onie a ABSTRACT .- B . G amsch and D . Lay [2] ha e s udied spec al mapping heo ems o he essen ial spec a o an ope a o ac ing in a complex Banachspace . Fi s ly hey conside ope a o sbelonging o he Banach algeb a o all bounded linea ope a o s on he space, and la e hey de i e he heo ems o unbounded closed linea ope a o s wi h non-emp y esol en om he aboye case ; bu boundedclosed linea ope a o s wi hdomain a p ope subspace a e no included . In his no e we in oduce a no ion o ex ended essen ial spec a o any closed linea ope a o wi hnon-emp y esol en , which co e s he aboye cases . Then, in his mo egene al con ex , we a e able o p o e he spec al mapping heo ems by me- ans o a mo euni ied app oach based on a ac o i- za ion o he ope a o s p o ided by he Dun o d- Taylo calculus and well-known p ope ies o p o- duc s o ope a o sp esen in F edholm heo y . AMS Subjec Classi ica ion (1980) : P ima y 47A60 ; seconda y 47A53 . Le X be a complex Banach space, C(X) he se o all closed linea ope a o s in  X , L(X) := {TE C(X)jD(T) = X} , T E C(X)  ;  D(T),  N(T),  R(T),  a(T),  B(T),  i(T),  a(T)  and  d(T) will deno e he domain, ke nel, ange, nulli y, de ec , ascen and descen o T , espec i ely . We shall conside he ollo- wingope a o classes : 0:= {T6 C(X)¡a(T) = B(T) = 0} 01 " =  {T E C(X)Ia(T),  B(T)  < -} m2 := {Te C(X)Ia(T) < - , R(T) complemen ed} m3:=  {T E C(X)1B(T)  < - ,  N(T)  complemen ed} 04 := {Te C(X)jR(T)  closed, a(T) < m} (D5 :=  {T E C(X)IB(T)  <  m} (D6'  (D 4U '5 (D 7  {T E C(X)Ia(T)  =  B(T)  <  m} (D A := {TI 1 7 ja(T) = d(T) < m} m9 :=  {T E C(X)Ia(T),  d(T)  <  m} (D10 := (T C(X)jR(T) closed} The essen ial spec a ai (T)  0,1, . .,10, a e de ined in e mso he abo e classes : ai(T) := {XECIXI-T~o i }, P i (T) := C-a i(T) , i = 0,1, . .10 ; no e ha p 0(T) := p(T) and a0(T) := a(T) , he usual esol- en and spec um . We now in oduce he ex ended essen ialspec a . De ini ion 106 Le  TEC(X) wi h p (T) qé ó and  aep(T) . Fo i= 0,1, . .,10, we de ine lo ¡ (T)  i  (a-T)-1E mi a ie (T) := a i (T)V {m}  o he wise . Thisde ini ion does no depend o a Ep(T)  because  - Ea ¡e (T) i  and only i e e y  S = L(X)  such ha  N(S)  =  {0}  and R(S) = D(T) does no belong o mi . No ice ha a oe (T) is he usual ex ended spec um a e (T) . F om now we conside  T E C(X)  wi h  p(T)  and aQp(T) . Le C be he ex endedcomplex plane and le A(T) be he se o all unc ions  : C ~> C  wi h domain an open se o( )  such ha  a e (T)C e( )  and   holomo phic on  e( ) . When CA(T) we conside wo open se s o, o'  such ha á ( )  =  o U o'  ,  o n o'  _ 0  ,   is  iden ically  0  on  o'  and is no iden ically 0 on each connec ed componen o e ; hen a e(T)  = o U o'  whe e  a' := a e (T) l1 0'  and  a := a e (T) o'  . Mo- eo e , E a will deno e he p ojec o associa edwi h he unc ion  e E A(T)  such ha  e(o) ={1} and  e(n') = {0}  . Lemma 1 " Le iE{0,1, . .,9} and xEC . Then : (a) >,-TE (D i  i and only i  (a-T) (a-T) -1 E o i (b)  E a E (D i  i  and only  i  a' no ¡e (T)  is emp y  " . P oo .  (a)  As  R[(a-T) -n ]  = D(T n )  we ha e  R[(a-T )n]  _ = R[((x-T)(a-T)-1)n] . On he o he hand, since (x-T)n(a-T)-nx = (a-T)-n(a-T)nx  o  x E D(T n )  and  (a-T) -n  is injec i e, we ha e N [D,-T) n 1 = N[((a-T)(a-T) -1 ) n ] . Now he esul is clea . (b)  We  ha e  a(E a )  =  d(E a )  <  1  ,  a(E a )  =  8(E a )  and R(E a ) complemen ed because E a is a p ojec o . Consequen ly, i i = 0 i is clea ha E a E m 0 i and only i o' = a'^ae(T) is emp y ; and o  i q£ 0 we ha e  E a E Oi  i and only i a(E 0 ) < -, ha is, i and only i o' is a ini e se whose elemen s a e poles o (a-T) -1 o ini e ank, o equi alen- ly a'n a ie is emp y . As he ze os o  E A(T)  a e isola ed poin s in  e  and a is compac , he e is only a ini e numbe o hem in a , say c0 = -,Cl,--,Ck wi h ini eo de s m 0 ?0, mi>0, i= 1, . .k . k mi Le m := m 0 +m l+ . .+m k and P(z) := n (c i -z) i=1 Lemma 2 " Le  E A(T) . Then : (a)  I  T e L(X)  ,  (T)  = F(T)P(T)E  whe e  F(z)  is  lo- a cally holomo phic in e( ) wi h no ze os in a e (T) . (b) I T L(X) , (T) = F a (T)P(T)(a-T) -m E a whe e F a (z) is locallyholomo phic in e( ) wi h no ze os in a e (T) " . P oo . De ine F(z) :=1  i z""  1  i zEe' and F  (z) := ~ (z)P(z)-1  i  z E e  a  1 (z)P(z) -1 (a-z i zEe . Now he esul is e iden om he p ope ies o he Dun o d Taylo calculus, [5J .  # Rema k Since F(z) and Fa (z) na e no ze os in a e (T) , he ope a o s F(T), F a (T)E L(X) a e in e ibles in L(X) . Mo e- o e ,  i  T E L(X)  ,  hen  (T)  can be exp essed as he p o duc o he commu ing ope a o s  F(T), E a , c i -T E L(X)  , i = 1, . .,k  ;  when  T O .L(X)  ,  hen  (T)  is exp essed as he p oduc o he commu ing ope a o s F , (T), E a , (a -T) -1 , (ci T)(a-T) 1 ELOC), i = l, ,k Lemma 3 Lemma 4 "  Le  T E L(X)  ,  E A(T)  and  j  =  1, . .,k  .  Then : (a) (T)E ~i i and only i c j -T, E6 Em i o i = 0,1, . .,10 . i = 0,1,2,3,4,5,8 . (b)  I  (T) E 0 6 we ha e  c j -T,  Ea ED 6  . (c) I  c j -T, EQ EcDi  hen  (T) E oi , o  i = 7,9 " . P oo .  Fi s ly we no e ha  F(T),  Fa (T)E (D . 1  o (a) The esul ollows om he ollowing : Le  A,B E L(X) wi hAB = BA ; hen AB EI>i i and only i A,B EO i . These a e well-known esul s ; see, o example, [3 ; 5 .3 .1] o i = 1,2,3 ; [1 ;(1 .3 .3),(1 .3 .4),(1 .3 .5)] o i = 4,5 ; [1 ; (1 .4 .8)] and [4 ;P op .9] o i=8 . (b)  I is consequence o  (a)  and he equali y  o6 = (D 4U 0 5" (c) Fo i = 7 i ollows om he addi i i y o he ín- dex o he p oduc o F edholmope a o s . On he o he hand, we know ha A,B E L(X)nO 9 and AB = BA imply AB Em 9 ,  [2 ;lemma 5] ; now, he esul ollows o i = 9 . # "  Le  T j L(X)  ,  EA(T)  and  j  =  1, . .,k  .  Then : (a) Fo i = 0,1,2,3,4,5,8we ha e (T) ,_ oi i and only i (cj-T)(a-T)-1,E F-Oil and (a-T) -1 Em i i m 0 ~ 0 . (b)  I  (T)F~1 6  we ha e  (cj-T)(a-T)-1,E a E m6,  and (a-T)-le o6 i m 0 YÉ 0 . (c)  I  (cj-T)(a-T) -1 ,E a P- Oi ,  and  (a-T) -1 E oi i m 0 ~ 0, hen (T)E(D i o i = 7,9 " . P oo . Imi a e he p oo o he lemma 3 .  # We now showspec al mapping heo em o essen ial spec a . Theo em " Le £A(T) . The ollowings a emen hold : (a) a l [ (T)] = [a ie (T)], i  0,1,2,3,4,5,8, (b) a6( (T)) :> [a6e(T)] . (c) a i ( (T))G [a ie (T)] . i = 7,9 P oo .  Gi en  N EC  ,  le  g (z) : =  N- (z)  .  Then  N ~ ai ( (T) ) i and only i g(T) E ~ . , i = 0,1, . .,9 . i (a)  F om lemmas 1,  3 and 4 we de i e ha  g(T) E 0 i  i and only i  g(z)  ~ 0  o e e y  z la ie (T)  ,  ha is,  i and only i VI [a ¡e (T)]  . No ice ha lemma 2 is applied o he unc ion gEA(T) . (b) and (c) a e also ob ained om lemmas 1, 3 and 4 in a simila way . Re e ences [2] GRAMSCH, B . and LAY, D .C . : "Spec al mapping heo ems o essen ial spec a" .- Ma h . Ann . 192 (1971), 17-32 . [4] SCHAEFER, H .H . : "On he F edholm al e na i e in locally con ex spaces" .- S udiaMa h . 18 (1959), 229-245 . Rebu el 16 de geneA del 1985 Uni e sidad de San ande Facul ad de Ciencias San ande SPAIN CARADUS, S .R ., PFAFFENBERGER, W .E . and YOOD, B . : "Calkin algeb as and algeb as o ope a o s on Banach spaces" .- Ma cel Dekke , 1974 . PIETSCH, A . : "Zu Theo ee de a-T ans o ma ionen in loka_1 kon exen Vek o áumen" .- Ma h .Nach . 21 (1960), 347-369 . TAYLOR, A .E . and LAY, D .C . : "In oduc ion o Func ional Analysis" .- Wiley 1980 .