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
.