Pub
.
Ma
.
UAB
Vol
.
30
Nó
1
Maig
1986
STATE
DIAGRAM
FOR
OPERATORS
6VITH
NULLSPACEOR
CONULL
SPACE
IN
AN IDEAL
OF
BANACHSPACES
Te esaAl a ez
1
.-
In oduc ion
Le
B be
he
class
o
all
Banach spaces
;
he
scala
ield
K is
ei he
he
eal
ield
o
he
complex
ield
.
Al1
ope a o s
ac ing
be ween
Banach
spaces
which
appea
in
his
a icle
a e
supposed
o be
linea
.
Fo
X,Y
e
B,£(X,Y)
is
he
space
o
all
ope a o s
om
X
in o
Y,
he
class
o
all
ope a o s
om
X
in o
Y
wi h
dense
domain
is
deno ed
by
cC
D
(X,Y),
IX
deno es
he
iden i y
ope a o on
X,
J
X is
he
embeddingmap
o
X
in o XII,
and
X
C
Y

means
ha
X is a
quo ien
space
o
Y
.
Fo
q
T
E
.C(X,Y),
D(T),
N(T)
and
R(T)
will deno e
he
domain,
null
space
and
ange
o
T
espec i ely,
and
we
also
w i e
CON(T)
:
=
Y/R(T),
CN(T)
:
=
Y/7),
while
a(T),
S(T)
and
j(T)
will
deno e he
dimension
o
N(T),
CON(T) and CON(T)
espec i ely
.
We
shall
conside
J6j5(X,Y)
:
=
{T
e
oC(X,Y)
:
T
is
no mally
sol able}
L(X,Y)
:
=

(T
E
£(X,Y)
:
T
is bounded)
Le
A
be an
ideal
o
Banach
spaces
.
Fo
in o ma ions
and
no-
a ions
abou ope a o
ideals
and
space
ideals
we
e e
o [51
.
We con-
side
he
ideals,
S,
R
o
F,
he
ideals o
all
sepa able,
e lexi e
o
ini e
dimensional
Banach
spaces
espec i ely
.
Some
no a ions
will
be
used
wi hou
explana ion
because
hei
meaning
is
ob ious
.
In
his
pape
we
ob ain
a
s a e
diag am
o
a
linea
ope a o
wi h
dense
domain
be ween
Banach
spaces
and i s
conjuga e
ope a o ,
and
we
p o e
ha his
diag am
is
comple e
.
2
.
"GENERALIZED"
CLASS
IFICATION
OF
(T,T')
:
STAT
E
DIAG
RAM
2
.1
.

THEOREM
.

Le
A be an
ideal
and
T
E
-L
D
(X,Y)
.
Then
:
(i)

U(V)
=
g(T),
a
(T)
_<
B(T')
;
in
gene al
he
inequali y
is
s ic
.

I , in
addi ion
T
£
JTs,
hen
a
(T)
=
d(T')
.
(ii)

Le
A
be
a
comple ely
symme ic
ideal,
hen
:
CON(T)
=
Res)
°=
N(T'),
86
(ii1
)
N(T')(EA
i
and
only i
CON(T)EA
(ii~
TIE
e $
:
N(T)
£A
i
and
only
i
CON
(T'
)C
A
.
(ii3
)
Suppose
A
su jec i e,
i
E0N(T')F-
A
hen
N(T)F-
A
.
Fo
a bi a y
ideals,
he
p ope ies
a e
no
alid,
in
gene al
.
P oo
.
(i)
I
is an
ob ious
consequence
o
he
duali y
ela ions,
N(T)'
=
X'/N(T)
0
=
(X'/R
T'))/(N(T)°/R
T'))
C
C0N(T')
To
see
ha ,
in
gene al,
he
inequali y
a
(T)
S
j(V)
is
s ic
we
de ine
TeL(1)

by
T(a
1
.

a2,
. .
.
.
Un
. .
.
.)

_

(0,

a
l
.

2
1 a2
,
.
.
. .
n-1
an

.)
.
conjuga e
ope a o
is T'(
0
n
)
=
(n
-1
9n+l
)'

(
0n)F-lm
(

U
n
)e
1
.
.I s
I is
ob ious
ha
N
(T)
= (0), 1_
/c,
q
C
CN(T')since
R(T')C
c,
.
Clea ly
C0N(T') $
R
since
i
m/c,E
R
henc,
has
,
a
subspace
isomo phic
o
l
m
;
ha
con adic s
c,e
S

and
1 <` S
.
(ii
1
)

I is
su ices
o
no ice ha
N(T')
=
COÑ
(T)'
and
. ha
-
Ais
comple ely
symme ic
.
(ii
2
)

I
Te
JdS
hen
N(T)'

=
CON
(T')
.
(ii
3
)
No e ha
N(T)'
q
C
CN(T'),
A
su jec i e
and
comple ely
symme ic
.
Fo a bi a y
ideals,
he
abo e
esul s
a e no
gua an eed
;
o
example,

i D
:
= (X
FE
B
:
J
XX is
complemen ed
in
X''}

,
Tl
,T
2
he
null
maps
on
1,
c,
espec i ely,
hen
N(T1)
=
l
m
4
S,
CON(T
1
)
=
le
S
N(TZ)
=
1(ED,
CON(T
2
)
= c,4-
D,
N(Tl)
=
1F-S
,
CON(T1)
=
l
m
4
S,
N(T
2
)
=
C
.4-D,
CQN(T2)
=
laD
.
We
now
in oduce
he
ollowing
classi ica ion
o
TC
.£
(X
.Y)
.
I

:

a
(T)
<

-
.
II

:
a(T)
_-
and
N(T)E
A
.
III

:
a
(T)
=-and
N(T)O A
.
1
:
B(T)< -
.
2
:
S(T)
=
mand
-Ñ(T)EA
.
3
:
B(T)
=-and
CON(T)$A
.
By
combining
hese
possibili ies
si ua ions
.
This
classi ica ion
scheme
may
conjuga e
T'
o T
.
The
p ope ies
o
he
(2
.1)
heo em
on he
language
o
he
p e ious
classi ica ioncan
be
w i en
as
:
T'EI~
TE1
T
< I
T'< 1
TE9$
:
Te
I
4=
:>
T'£
1
A
comple ely
symme ic
:
T'EIII41
:
:~
TE3
A
comple ely
A
comple ely
symme ic
and
TE,"
:
TEIII4=J
T'£3
.
We
shall
p oceed
o
cons uc
a
diag am
.
The
shaded
in
he
diag am
co espond
o
s a es ha
a e
imposible
by i ueo
(2
.1)
heo em
.
III3
III
2
III
1
II
3
II
2
1
2
symme ic
and
su jec i e
:
T
E
III
==4>
TIC
3
1
213

II1

II
2

II3

1111

1112

1113
we
ob ainnine
di e en
now
be
applied
o
he
squa es
~~II
.uS
~S
11
cs
~~
~QS
.Ñ$ ~uS
-'~
-
~
.
-"'
FIEN
-"
mal
cNS'
,NS
s1~
1
,
9,
1
»
x@
We
analyse
i
he
diag am
is
comple e,
so,
we
p o e
ha
a
p ocedu e
o
cons uc
s a e
exampleso
L(X,Y)
in
he
Taylo -Haldbe g
classi ica ion,
in oduced
in
1962
by
Goldbe g
and
Tho p
[2], is
alid
o ou
classi ica ion
.
I El
,
E2
E
B
hen
he map
A

:
h
E
(
El xE2
)'

!>(hl, h
2
)FE
i
x Ez
whe e
h
1
(x
1
) :
=
h(x
l
,
0),
h2 (x
2
)
:
=
h(0,

x
2
),

xi
e
E
i
,

i
=
1,2
is
an
isomo phic,

hus
we
can
iden i y
El
x
E2 e (E
l
x E2
)'
h ough
he
o mula
($)

(h
l
,
h
2
)
(xl
,
x2
):
=
h(x
l
,
x
2
):
= h
l
(x
1
)
+
h
2
(x2
)E
K
whe e
(x
1
,

x
2
)E
El x E2
.
Fo
T1
E
oC
D
(X
1'
Y1
)'
T2
E
aC
D
(X
2
'
Y
2
)
i is
possibleiden i y
Ti
x T2 o
(T
1 x T
2 )'

by
using
he
($)
o mula
o
conside
(Ti h
1
,
T2
h
2
)
as
an
elemen
o
(X
1
x X2
)'
.

Also i is
clea
ha
i
we
de ine
he
p oduc
be ween
wo
s a eso
ou classi ica ion
by
using
he
o mula
(Aa
,
Bb
)
x
(C
c
,

Dd
)
:
=

(max(A,C)
max(a,c)'
ax(B
'
D)
max(b,d)
)
he
s a e
o
he
ope a o
T
1
x T2 is he
p oduc
o
he
T1
and
T
2
s a es
.
(2
.2)
THEOREM
.
The
s a e
diag am
o
(T,T')
is
comple e
.
P oo
es

:
Impossible
i
A
is
comple ely
symme ic
es
+05

:
Impossible
i A is
comple ely
symme ic
and
T
E
-WS
es
+
s

:
Impossible
i
A
is
comple ely
symme ic
and
su jec i e
.
,g$

:
Impossible
i
T
E
.NS
(I
1
,
1 1
)
:
Le
T
be
he
iden i y
ope a o
in X
.
(I
1
,
I
2
)
:
Le
A =
R,

(x
i
)
i6
I

a
no malized
Hamel
basis
o
1
2
(N),
88
(e
i )
i
E
I
an
o hono malbasis
o
12
(I)
.
De ine
T
:

D(T)C1
2
(I)
---i
1
2
(N)
e
l
-STe
i
:
=,x
i
whe e
D(T)
is he
linea span o
he
e,'s
.
Clea ly
D(T)
is
dense

in

12
(1),

11(1')

=

1 2
(N)

and

N(T)

=

{O
) .
Le
(enk
)kE
N
C
(e
i)
i
El
be a
sequenceo
di e en
ec o s
and
xm
:
=
kil
e
n
/k
2
ED(T),
mEN
.
Then
xm-->
xn=kElen
/k2,
k

k
Txm
-j
yn
:
= k
E l

xn /k2
E
R(T)
i m

hence he e
k
exis s
zn
E
D(T)
such ha Tz
n= y
n
,
mo eo e
x
n
-z
n
~ 0
since
xno
D(T),
Consequen ly
o y
E
D(T')
we
ha e
ha
<xn - zn
,
T'y
> = 0
hus
xn -
z
n
E
R(V)°
.
We
can
choose
(e
n
)

C
(ei)iE
.I
disjoin
sequences,
k"
k
EN
hus
o
n
E
Nwe
ob ain
(xn -
zn)
nE
Ñ
R(T')°
;
mo eo e ,
x
n - zn
a e
linea ly
independen ,
hence
dim
R
T'

°
=
m
.
Clea ly
1
2
(I)/R
T'

E
R
.
(1 1'
1
3
)

Le
A
= R
and
T
be
he
ope a o
in
(2
.1)
heo em
(1
2 ,11
1
)

:

Le
A be
comple ely
symme ic,
XEF,
YEA
-
F,
T' bé
.
null
map
om
X
in o
Y
.
(1
2
,111
1
) :

Le
A
be
non
comple ely
symme ic,
XEF,
YEA,
Y'4-:A,
T
he
null
map
om
X
in o
Y
(1 3' 11
1
)

Le
A
be
non
comple elysymme ic,
XEF,
Y
4
A,
Y'E
A,
T
he
null
map
om
X
in o
Y
(1
3
,111
1
)

:

Le
Abe
comple ely
symme ic,
M
C
X

A,
MEA,
X/M
é
.A,
T
he
inclusion
om
X
in o
Y
(11
1 ,1
2
) :

In
he
example
(1
2
,11
1
)
i
su ices
o
eplace
T
by
he
conjuga e
ope a o
.
(111
1,1
2
)
:

In
he
example
(1
2 ,111
1
)
i
su ices
o
eplace
T by
he
conjuga e
ope a o
(III
1
,1
3
)'

In
he
example
(1
3 ,111
1
)
i
su ices
o
eplace
T by
he
conjuga e
ope a o
.
We
can
ob ain
he
emaining
allowed
s a esby
applica ion
o
he
p e ious
p ocedu e
.

S
.GOLDBERG
.
Unbounded
linea
ope a o s
.
Mc
G aw-Hill,(1966)
.
S
.
GOLDBERG,
E
.0
.
THORP
.
The
ange
as
angespace
o
compac
ope a o s
.
J
.
Reine
Angew
.
Ma h
.,
211,
(1962),
113-115
G
.J .O
.JAMESON
.
Topolog
y
and no med
spaces
.
Chapman
and
Hall,
(1974)
.
J
.
LINDENSTRAUSS,
L
.
TZAFRIRI
.
Classical
Banach spaces
I,
Sp inge -Ve lag,
(1977)
.
A
.
PIETSCH
.
Ope a o
ideals
.
No h-Holland,
(1980)
.
[6]

A .E
.
TAYLOR

C
.J
.A
.
HALBERG
.
Gene a
l
heo ems
abou
a
bounded
linea
ope a o
and i s
conjuga e,
J
.
Reine
Angew,
Ma h
.,
198,
(1957),
93-111
.
Rebu
el
14
de
no embne
del
1985
Te esa
Al a ez
Depa amen o
de
Teo ía de
Funciones
Facul ad
de
Ciencias
Uni e sidad
de
San ande
San ande ,
ESPÁÑA
REFERENCES