Appl
.Mat
h.J.Chi
ne
s
eUni
v.Se
r
.B
2006,21(1):6468
PROPERTI
ESOFpIJKPOLORMNOOPERNTORS
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RS
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m
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eR ekQ
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h
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t^l
65
PROPERTI
EsOlpωHYPONORMApOPERATORs
t
hec
l
as
sofl
oghyponor
malope
r
at
or
sc
ont
ai
nst
heot
he
ronei
n[5].Theaut
hors
howe
d
,buti
t
hati
fT i
saTanahas
hiope
r
at
oronH,t
he
nT⊕0onH⊕H i
sωhyponor
mal
ti
snot
phyponor
malorl
oghyponor
mali
n [4].Thus t
he c
l
as
s ofωhyponor
malope
r
at
or
s
.Put
pr
ope
r
l
yc
ont
ai
nst
hec
l
as
s
e
sofphyponor
maland l
oghyponor
malope
r
at
or
s
nam
pr
ove
dt
hes
pe
c
t
r
um pr
ope
r
t
i
e
sofhyponor
malope
r
at
or
si
n[6].Re
c
e
nt
l
y,i
n[7]and[8],
.I
t
hepr
ope
r
t
i
e
shavebe
e
ne
xt
e
nde
dt
ophyponor
malope
r
at
or
s
n[9]and[10],Al
ut
hge
.I
,wes
hase
xt
e
nde
dt
he
mt
oωhyponor
malope
r
at
or
s
nt
hi
spape
r
hal
le
xt
e
ndt
he
mt
opω-
hyponor
malope
r
at
or
sandc
ons
i
de
rt
her
e
l
at
i
onoft
hes
pe
c
t
r
aandnume
r
i
c
alr
angeofpω-
.Att
hyponor
malope
r
at
or
s
hee
nd,f
ort
hepωhyponor
malope
r
at
orwes
hal
lpr
ovet
hati
f
～
～
,t
Ti
snor
mal
he
nT=T.
PQ RST
UV
W
X
YZ
[
X
T‘oft
he
Ac
ompl
e
xnumbe
r\i
ss
ai
dt
obei
nt
heappr
oxi
mat
epoi
nts
pe
c
t
r
um ]
^_
ope
r
at
orT i
ft
he
r
ei
sas
e
aue
nc
ebcdeofuni
tve
c
t
or
ss
at
i
s
f
yi
ng_Tf \‘cdg 0,andi
fi
n
h
i
‘cg0,t
addi
t
i
on_T f\
he
n\i
ss
ai
dt
obei
nt
hej
oi
ntappr
oxi
mat
epoi
nts
pe
c
t
r
um ] _T‘
d
k
^
oft
heope
r
at
orT.Theboundar
yoft
hes
pe
c
t
r
um ofanyope
r
at
orT i
sas
ubs
e
toft
he
appr
oxi
mat
epoi
nts
pe
c
t
r
um ]^_T‘oft
heope
r
at
orT.loranybounde
dl
i
ne
arope
r
at
or
sm
.pe
andT,i
ti
snnownt
hatt
henono
e
r
opoi
nt
sof]_mT‘and]_Tm‘ar
ei
de
nt
i
c
al
tT=qrTr
h
h
h
bet
hepol
arde
c
ompos
i
t
i
onofT.si
nc
erT r= qrTrq ,rTr= q qrTr,t
henono
e
r
o
h
.
poi
nt
sof]_rT r‘and]_rTr‘ar
ei
de
nt
i
c
al
[9]
,\
wx.I
tW
uuSQ.v. pe
tTbes
e
mi
hyponor
mal
ft
hes
e
aue
nc
ebcdeofuni
tve
c
t
or
si
ss
uc
h
h
i
‘cdg0,t
‘cdg0.
t
hat_Tf\
he
n_T f\
[11]
,\y 0.I
tW
uuSQ.Q. pe
tT beωhyponor
mal
ft
hes
e
aue
nc
ebcdeofuni
tve
c
t
or
si
ss
uc
h
h
h
i
‘cdg0,t
r‘cdg 0.I
‘cd
he
n_rT rfr\
fi
naddi
t
i
onT i
si
nve
r
t
i
bl
e,t
he
n_T f \
t
hat_Tf\
g0.
,\y 0.I
z{W
|V
W
u Q.}.pe
tT bei
nve
r
t
i
bl
epωhyponor
mal
ft
hes
e
aue
nc
ebcdeofuni
t
h
i
‘cdg0,t
‘cdg0.
ve
c
t
or
sar
es
uc
ht
hat_Tf\
he
n_T f\
p
～p
～h p
～
p
.lorap~V
||f
ωhyponor
malope
r
at
orT,wege
trTr ≥ rTr ≥ rT r.ThusT i
s 2
～
.
hyponor
malandT i
sωhyponor
mal
‘cd‖≥r\
rf‖Tcd‖,wi
si
nc
e‖_Tf\
t
houtl
os
sofge
ne
r
al
i
t
y,weas
s
umeTy0.pe
t
1
b‖ Tcd‖ e= b‖ rTrcd‖ ebebounde
dawayf
r
om 0.pe
tyd= rTr2cd,t
he
nb‖ yd‖ ei
s
bounde
dawayf
r
om 0.
1
1
～
～f ‘ g 0.
,_T
si
nc
e_Tf \‘cdg 0,t
he
n_TrTr2f rTr2\‘cdg 0,t
hati
s
\ yd
Byus
eof
～h i
～p
～h rpfrrp‘ g0.
‘ydg0,s
rp‘ydg0,and_rT
pe
mma2.2,_T f\
o_rTrfr\
\ yd
～p
～h p
p
lorpωhyponor
malope
r
at
orT,wege
trTr≥rTr≥rT r,t
he
n
～h p
～p
～h p
p
p
0≤ __rTr f rT r‘yd,yd‘≤ __rTr f r\
r‘yd,yd‘f __rT r f r\
rp‘yd,yd‘g 0.
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
66
.21,No.1
Vol
～* p
p
He
nc
e(|T|-|T |)yn→0,and
～
～
(|T|p - |λ
|p)yn = (|T|p - |T* |p)yn + (|T* |p - |λ
|p)yn → 0.
1
p
p
|)xn→0.
Thus|T|2(|T|-|λ
p
|p)xn→0.ThusU|T|1- p(|T|p-|λ
|p)xn→0,t
Si
nc
eT i
si
nve
r
t
i
bl
e,t
he
n(|T|-|λ
hat
|p)xn→0,s
-|λ
|pU|T|1- p)xn→0.
,(T-U|T|1- p|λ
o(λ
i
s
*
p- 1 - i
θ *
1- p
-i
θ
,U* xn- |λ|p- 1e
|T|1- pxn→ 0,
Be
c
aus
eU xn- |λ| e U U|T| xn→ 0,t
hati
s
* p
-i
θ
|pxn→0.ThusU|T|U* xn-|λ
|xn→0,t
,TU* xn-e
hati
s
Txn→0,
t
he
n(U|T|U )xn-|λ
*
-i
θ
*
-i
θ
*
nc
eT i
si
nve
r
t
i
bl
e,t
he
n(U -e )xn→0.ForT=U|T|,U T=
andT(U -e )xn→0.Si
*
*
*
*
*
U U|T|=|T|.ThusT xn=|T|U xn=U TU xn.
*
- 2i
θ
-i
θ
*
,(T* -|λ
|e
)xn→0,t
)xn→0.
Be
c
aus
eT xn-e λ
xn→0,t
hati
s
hus(T -λ
,t
T)-0.
Cor
ol
l
ar
y2.4.Le
tT bei
nve
r
t
i
bl
epωhyponor
mal
he
nσja(T)-0=σ
a(
Le
mma 2.5.Le
tT= U|T|bet
hepol
arde
c
ompos
i
t
i
onofT,λ∈ C,{xn}i
st
heve
c
t
or
*
*
)xn→0and(T -λ
)xn→ 0,t
|)xn→ 0and(|T |- |λ
|)xn
s
e
que
nc
e.I
f(T-λ
he
n(|T|- |λ
-i
θ
i
θ
*
-i
θ
→0.I
=|λ
|e
≠0,t
fλ
he
n(U-e)xn→0,(U -e )xn→0.
*
.Foranyx∈H,‖Tx‖=‖|T|x‖,t
Pr
oof
he
nTxn→0⇔|T|xn→0.Si
mi
l
ar
l
y,T xn→0⇔
|T* |xn→0,t
,λ
=0,t
hati
s
hi
sr
e
s
ul
ti
st
r
ue.
-i
θ
=|λ
|e
≠0,t
|and|T* |+|λ
|ar
Le
tλ
he
npos
i
t
i
veope
r
t
or
s|T|+|λ
ei
nve
r
t
i
bl
e,
*
*
(|T|+ |λ
|)(|T|- |λ
|)= T (T- λ
)+ λ
(T - λ
),
*
*
*
(|T |+ |λ
|)(|T |- |λ
|)= T(T - λ
)+ λ
(T- λ
).
*
*
)xn→0,t
|)xn→0,(|T |-|λ
|)xn→0,
)xn→0,(T -λ
hus(|T|-|λ
So(T-λ
i
θ
|λ
|(U- e
)= (T- λ
)- U(|T|- |λ
|),
*
-i
θ
*
*
*
|λ
|(U - e )= (T - λ
)- U (|T |- |λ
|).
i
θ
*
(1)
(2)
(3)
(4)
-i
θ
The
n(U-e)xn→0,(U -e )xn→0.
i
θ
,λ= |λ|e ≠ 0,{xn}i
st
heve
c
t
or
Cor
ol
l
ar
y2.6.Le
tT bei
nve
r
t
i
bl
epωhyponor
mal
i
θ
)xn→0,t
|)xn→0and(|T* |-|λ
|)xn→0;So(U-e
)xn→0,
s
e
que
nc
e,(T-λ
he
n(|T|-|λ
-i
θ
(U* -e
)xn→0.
:
Re
c
al
lt
henume
r
i
c
alr
angeW(T)ofT wi
t
hi
t
sde
f
i
ni
t
i
onasf
ol
l
ows
W(T)= {(Tx,x):x∈ H,‖x‖ = 1}.
Le
tW (T)bet
hec
l
os
ur
eofW(T).Foranyope
r
at
orT,asweal
lknow,W(T)i
sac
onve
x
,t
s
e
tandσ(T)⊆W (T).I
fTi
snor
mal
he
nW (T)=c
onv(σ(T)),whe
r
ec
onv(σ(T))me
ans
t
hec
onve
xhul
lσ(T).
1
1
～
[9]
Le
mma2.7 .Le
tT=U|T|bet
hepol
arde
c
ompos
i
t
i
onofT.T=|T|2U|T|2,t
he
nσ(T)
～
=σ(T).
[9]
- ～
- ～*
,t
Pr
opos
i
t
i
on2.8 .Le
tT beωhyponor
mal
he
nW (|T|)⊆W (|T |).
～|)⊆W
～* |).
-(|T
-(|T
,t
The
or
e
m 2.9.Le
tT bepωhyponor
mal
he
nW
～p
～* p
～ ～～
p
.ForapPr
oof
ωhyponor
malope
r
at
orT,wehave|T|≥|T|≥ |T |.Le
tT=U|T|be
～
～* ～
～
～*
～ ～ ～* ～ ～* ～ ～
t
hepol
arde
c
ompos
i
t
i
onofT,t
he
n|T |=U|T|U ,|T|=U U|T|,andσ(|T|)=σ(U U
.
YangChangs
e
n,e
tal
PROPERxr
ESOFpωHYPONORMALOPERAxORS
67
～|)= (～|～|～* )= (|～* |),
～
～*
～p
|T
σU T U
σ T
t
husσ(|T|)- {0}= σ(|T |)- {0}.Be
c
aus
eof|T|≥
～* |p,
～
～*
～
～*
～|)=c
～
-(|T
|T
t
he
n0∈σ(|T|),σ(|T |)andσ(|T|)⊆ σ(|T |),t
husW
onvσ(|T|)⊆
～*
- ～*
c
onvσ(|T |)=W (|T |).
～|p)⊆W
～* |p).
-(|T
-(|T
Re
mar
k.Wi
t
hc
ar
e
f
ulobs
e
r
vat
i
onweknow t
hatW
～
～
p
* p
,t
The
or
e
m 2.10.Le
tT bepωhyponor
mal
he
nσ(|T|)⊆W(|T |).
.ForapPr
oof
ωhyponor
malope
r
at
orT unde
ranyx∈H,
～p
～
p
(|T|x,x)≥ (|T|x,x)≥ (|T* |px,x),
(5)
～
～
～
p
p
* p
t
he
n(|T|x,x)∈W(|T|)⊆W (|T |).
～* p
～* |p).An
p
-(|T
t(|T|x,x)∈W
dhe
nc
e
Byus
eoft
hec
onve
xi
t
yofW(|T |)and(5)wege
～p
p
- ～p
- ～* p
σ(|T|)⊆c
onv(σ(|T|))=W (|T|)⊆W (|T |).
ef gohi
j
klm
hnr
e
mar
k
o
Forωhyponor
malope
r
at
or
swehaveknownt
hef
ol
l
owi
ngr
e
s
ul
t
s
～
*
*
,t
(p)Le
,qe
rT.r
fT i
snor
mal
he
n
tT beωhyponor
mal
rT⊆qe
rT orqe
rT ⊆ qe
～.(
T=T s
e
estu,vor
ol
l
ar
yt).
～
～
(w)Le
,i
,t
tT beωhyponor
mal
fT i
snor
mal
he
nT=T.(s
e
espwu,xhe
or
e
m p).
,and pr
We s
hal
le
xt
e
nd (w) t
o pωhyponor
malope
r
at
or
s as f
ol
l
ows
e
par
et
he
.
f
ol
l
owi
ngLe
mmasf
i
r
s
t
*
*
*
*
*
*
ye
mmaf.1.Le
tz=z ,{={ ,|(} )~|(z)and|(} )~|({).r
f} z}=} {},
t
he
nz={.
*
*
*
*
.Si
Pr
oof
nc
e} z}= } {},t
he
n(z}x,}x)= (} z}x,x)= (} {}x,x)= ({}x,
}x).Be
c
aus
ez|R(})={|R(}),s
oz|R(})={|R(}).
*
*
Si
nc
e|(} )~|(z),|(} )~|({),t
he
nz||(}* )={||(}* )=0.xhusz={.
p
p
w
p
p
w
ye
mmaf.2.Le
tz,{≥0.r
f{wz{w={ andzw{zw=z ,t
he
nz={.
.
Pr
oof
p
p
p
w
w
p
Si
nc
e{wz{w={ ,t
he
n(z{x,{x)= ({z{x,x)= ({w{ {wx,x)= ({{x,{x),t
hus
z|R({)= {|R({),andz|R({)= {|R({).Now f
oranyx∈ |({),be
c
aus
e({zx,{zx)= (x,
w
t
,{zx=0.
z{ zx)=(x,{ zx),t
he
n({zx,{zx)=0,t
hati
s
p
p
w
t
t
,x∈ |(z )= |(z),t
Si
nc
ezw{zw= z ,t
he
nz x= z{zx= 0,t
hati
s
he
n|({)~
|(z).Byus
eofLe
mmat.p,wehavez={.
～
～
,0<p≤p.r
,t
The
or
e
m f.f.Le
tT=U|T|bepωhyponor
mal
fT i
snor
mal
he
nT=T.
～
～* ～ ～～*
～|w=|～* |w.
.Si
,t
,|T
Pr
oof
nc
eT i
snor
mal
he
nT T=TT ,t
hati
s
T
Fort
hepωhyponor
mal
～
～
～
～
p
p
* p
*
ope
r
at
orT,|T|≥|T|≥|T |,t
he
n|T|=|T|=|T| .
p
p
～|=|
* w
*
*
(p)|T
T|i
fandonl
yi
f|T |=|T |w|T||T |w.
p
p
p
p
*
w
*
～
,s
r
nf
ac
t
i
nc
e|T|= |T|,t
he
n|T|wU |T|U|T|w= |T|,t
he
nU|T|wU |T|U|T|w
p
p
*
w *
,|T* |w=|T* |w|T||T* |w.
U =U|T|U ,t
hati
s
68
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
.21,No.1
Vol
.
Whatwi
l
lbeont
hec
ont
r
ar
y,i
ti
sde
arandobvi
ous
1
1
～*
2
*
(2)Si
mi
l
ar
l
y,|T |=|T|i
fandonl
yi
f|T|=|T|2|T ||T|2.
1
1
1
1
～
～*
*
*
*
Si
nc
e|T|=|T|=|T |,t
he
n|T |2|T||T |2=|T|2|T ||T|2.Byus
eofLe
mma
～.
3.2,(1)and(2),wehave|T* |=|T|and|T|U=U|T|,he
r
c
eT=T
Re
f
e
r
e
nc
e
s
1 Al
ut
hgeA.Onphyponor
ma
lope
r
a
t
or
s0<p≤1,I
nt
e
gr
a
lEqua
t
i
onsOpe
r
a
t
orThe
or
y,1990,13:307315.
2 Al
,I
10.
ut
hgeA,Wa
ngD.ωhyponor
ma
lope
r
a
t
or
s
nt
e
gr
a
lEqua
t
i
onsOpe
r
a
t
orThe
or
y,2000,36:13 Al
,1999,2:
ut
hgeA,Wa
ngD.Anope
r
a
t
ori
ne
qua
l
i
t
ywhi
c
hi
mpl
i
e
spa
r
a
nor
ma
l
i
t
y,Ma
t
hI
ne
qua
lAppl
113119.
4 Al
ut
hgeA,Wa
ngD.ωhyponor
ma
lope
r
a
t
or
sⅡ,I
nt
e
gr
a
lEqua
t
i
onsOpe
r
a
t
orThe
or
y,2000,37:324331.
5 Ta
,I
372.
na
ha
s
hiK.Onl
oghyponor
ma
lope
r
a
t
or
s
nt
e
gr
a
lEqua
t
i
onsOpe
r
e
a
t
orThe
or
y,1999,34:3646 Put
,Tr
,1974,188:
na
m CR.Spe
c
t
r
aofpol
a
rf
a
c
t
or
sofphyponor
ma
lope
r
a
t
or
s
a
nsAme
rMa
t
hSoc
419428.
7 ChoM,Hur
,Gl
uyaT,I
t
ohM.Spe
c
t
r
aofc
ompl
e
t
e
l
yphyponor
ma
lope
r
a
t
or
s
a
s
ni
kMa
t
h,1995,30:6167.
-M,I
8 Cho
,Pr
,1995,123:
t
ohM.Put
na
m'
si
ne
qua
l
i
t
yf
orphyponor
ma
lope
r
a
t
or
s
ocAme
rMa
t
hSoc
24352440.
9 Al
,Hokka
ut
hgeA,Wa
ng D.Put
na
m'
st
he
or
e
ms f
or ωhyponor
ma
l ope
r
a
t
or
s
i
do Ma
t
he
ma
t
i
c
a
l
,2000,29:383389.
J
our
na
l
10 Al
,Hokka
ut
hgeA,Wa
ngD.Thej
oi
nta
ppr
oxi
ma
t
epoi
nts
pe
c
t
r
um ofa
nope
r
a
t
or
i
doMa
t
he
ma
t
i
c
a
l
,2002,31:187197.
J
our
na
l
11 Xi
,Bos
aD.Spe
c
t
r
a
lThe
or
yofHyponor
ma
lOpe
r
a
t
or
s
t
on:Bi
r
kha
us
e
rVe
r
l
a
g,1983.
12 ChoM,Hur
,JofI
,2002,7(1):110.
uyaT,Ki
m Y O.A not
eonωhyponor
ma
lope
r
a
t
or
s
ne
qua
lAppl
.ofMa
De
pt
t
h.,He
na
nNor
ma
lUni
v.,Xi
nxi
a
ng453007,Chi
na.