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PureMathematicsnØêÆ,2022,12(10),1810-1825
PublishedOnlineOctober2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1210194
Banach˜mþ†ãL«
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«½Â,'uÂ:†u:‡¼f±9C¼f½Â,|^mN½n,“êÓ½n½
Â,`²üa‡¼fŒÏLC¼fïᘇª,,?؆ ãBanachL«ƒm gÓ
8†Ù3‡¼fŠ^eBanachL«ƒmgÓ8éA ‡¼fN“ ê5Ÿ,y²‡
¼fN´‡“êÓ"
'…c
Banach˜m§†ã§L«§‡¼f
TheReflectionFunctorsof
RepresentationsofQuiverson
BanachSpaces
JiahuaQue,YunnanZhang
SchoolofMathematicsandStatistics,FujianNormalUniversity,FuzhouFujian
Received:Sep.20
th
,2022;accepted:Oct.19
th
,2022;published:Oct.26
th
,2022
©ÙÚ^:UZu,ÜH.Banach˜mþ†ãL«‡¼f[J].nØêÆ,2022,12(10):1810-1825.
DOI:10.12677/pm.2022.1210194
UZu§ÜH
Abstract
ThepurposeofthispaperistoextendsomepropertiesofHilbertspacestoBanach
spaces.ThispapergavethedefinitionsofthequiversandtheirBanachrepresenta-
tions,thedefinitionsofthereflectionfunctorsatthesinksandthesources,andthe
definitionsofcontravariantfunctors.Byusingtheopenmappingtheorem,algebraic
isomorphismandotherdefinitiontheorems,itisshownthattwokindsofreflection
functorscanestablishanequalitythroughcovariantfunctors.Italsodiscussedthe
algebraic properties ofthereflectionfunctormapcorrespondingtotheautomorphism
setsbetweentheBanachrepresentationsofthequiversandtheirBanachrepresenta-
tionsundertheactionofthereflectionfunctors.Itprovedthatthereflectionfunctor
mapisanalgebraicisomorphism.
Keywords
BanachSpaces,Quivers,Representations,ReflectionFunctors
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
Γ´†ã,Gabriel([1])ïÄk•‘‚5˜mþ†ãL«,ͶGabriel½n:k•ë
φã=kk•õØŒL«¿‡^‡´Ù.Õã´DynkinãA
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†ãL«.[11]3Hilbert˜m‰ÆeïĆãÛÜXþL«.[12]ïÄá‘Hilbert˜mþ†
ãL«.Hilbert˜mþ†ãL«®²'¿©ïħ·‚òrHilbert˜mþ†ãL«
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†ã9ÙBanachL«½Â,'uÂ:†u:‡¼f±9C¼f½Â,¿`² üa‡¼
fŒÏLC¼fïᘇª,1n!?؆ãBanachL«ƒmgÓ8†Ù3‡¼fŠ^
eBanachL«ƒmgÓ8éA‡¼fN“ê5Ÿ,y²‡¼fN´‡“êÓ.
e¡‰Ñ˜½Â†PÒ.X´Banach˜m,±X
∗
L«Xݘm.X,Y´Banach˜
m,±B(X,Y)L«lXY¥k.‚5ŽfN,{PB(X,X)=B(X).XþðŽfP•I.
DOI:10.12677/pm.2022.12101941811nØêÆ
UZu§ÜH
T∈B(X,Y),±ker(T) ={x∈X:Tx=0}†Im(T)={Tx:x∈X}©OL«T"˜m†Š
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M
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i
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i∈J
X
i
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Uìþã‰ê¤•˜‡Banach˜m,Ùݘm•
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E•†Þ8.w,†ã´‡k•ã.
eV,EÑ´k•8,K¡†ãΓ•k•†ã.
½Â2.2Γ=(V,E,s,r)´†ã,eX=(X
v
)
v∈V
´Banach˜mx,f=(f
α
)
α∈E
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α
∈
B(X
s(α)
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v
=0,KPX=(X
v
)
v∈V
=0,dž¡BanachL«(X,f)•"L«,P
•(X,f) = 0.
e¡‰Ñ†ãBanachL«ƒmÓ†Óܤ½Â.
½Â2.3Γ=(V,E,s,r)´†ã,(X,f),(Y,g)´ΓBanachL«,T=(T
v
)
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v
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α
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α
T
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Hom((X,f),(Y,g)) = {l(X,f)(Y,g)Ó}.
gÓ8
End((X,f)) = Hom((X,f),(X,f)).
P"gÓ0= (0
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)
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,ðgÓI= (I
v
)
v∈V
.
DOI:10.12677/pm.2022.12101941812nØêÆ
UZu§ÜH
Ún2.4Γ = (V,E,s,r)´†ã,(X,f),(Y,g),(Z,h)´ΓBanachL«.
(1)T∈Hom((Y,g),(Z,h)),S∈Hom((X,f),(Y,g)),é?¿v∈V,P(TS)
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v
S
v
,
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v
)
v∈V
,¡•TÚSܤ,KTS∈Hom((X,f),(Z,h));
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v
= T
v
+S
v
, PT+S= ((T+S)
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)
v∈V
,
¡•TÚSÚ,KT+S∈Hom((X,f),(Y,g));
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v
=λT
v
,PλT=((λT)
v
)
v∈V
,¡
•Tê(λ)¦,KλT∈Hom((X,f),(Y,g)).
y²d½ÂŒ†y
w,†ãBanachL«UìþãÓ89Ùܤ¤˜‡‰Æ,=k
·K2.5Γ =(V,E,s,r)´†ã,KΓ¤kBanachL«Uì½Â2.3¥Ó8†Ún2.4¥
ܤ¤˜‡‰Æ,P•BRep(Γ).
e¡‰Ñ†ãBanachL«Ó!†Ú!ØŒ©)ÚŒD4½Â.
½Â2.6Γ = (V,E,s,r)´†ã.
(1)(X,f),(Y,g)´ΓBanachL«.e•3l(X,f)(Y,g)þÓT,=T=(T
v
)
v∈V
∈
Hom((X,f),(Y,g)),…é?¿v∈V,T
v
∈B(X
v
,Y
v
)´ÓŽf,K¡(X,f)†(Y,g)Ó,P
•(X,f)
∼
=
(Y,g).
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0
,g
0
)´ΓBanachL«.eé?¿v∈V,?¿α∈E,X
v
=Y
v
L
Y
0
v
,
f
α
=g
α
L
g
0
α
,KPX=Y
L
Y
0
,f=g
L
g
0
,dž¡(X,f)´(Y,g)†(Y
0
,g
0
)†Ú,P•(X,f)
= (Y
L
Y
0
,g
L
g
0
) = (Y,g)
L
(Y
0
,g
0
).
(3)(X,f)´Γš"BanachL«.e(X,f)ÓuΓü‡š"BanachL«†Ú,K
¡(X,f)´Œ©).ÄK¡(X,f)´ØŒ©).
e5‰Ñ†ãBanachL«'uÂ:†u:‡¼f±9C¼f½Â,¿`²üa‡
¼fŒÏLC¼fïᘇª.Äk‰Ñ†ã¥Â:†u:9ÙƒA)¤#†ã,±9é
ó†ã½Â.
½Â2.7 [12]Γ = (V,E,s,r)´†ã,v∈V´‡º:.
(1)eé?¿α∈E,s(α) 6= v,K¡v´ΓÂ:,džP
E
v
= {α∈E: r(α) = v}.
(2)eé?¿α∈E,r(α) 6= v,K¡v´Γu:,džP
E
v
= {α∈E: s(α) = v}.
F⊆V×V´‡†Þ8,α∈F.eα: x→y,Pα: y→x.PF= {α: α∈F}.
½Â2.8 [12]Γ = (V,E,s,r)´†ã.
DOI:10.12677/pm.2022.12101941813nØêÆ
UZu§ÜH
(1)v∈V´ΓÂ:,Pσ
+
v
(V) = V,σ
+
v
(E) = (E\E
v
)
S
E
v
,-†ã
σ
+
v
(Γ) = (σ
+
v
(V),σ
+
v
(E),s,r).
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−
v
(V) = V,σ
−
v
(E) = (E\E
v
)
S
E
v
,-†ã
σ
−
v
(Γ) = (σ
−
v
(V),σ
−
v
(E),s,r).
(3)PV= V,E= {α: α∈E},-Γéó†ã
Γ = (V,E,s,r).
e¡‰Ñ†ãBanachL«'uÂ:†u:‡¼f±9C¼f½Â.Äk‰Ñd
†ãΓBanachL«(X,f)'uÂ:vpσ
+
v
(Γ)BanachL«,(X,f)'uu:vpσ
−
v
(Γ)
BanachL«,±9(X,f)pΓBanachL«.
½Â2.9Γ = (V,E,s,r)´k•†ã.
(1)v∈V´ΓÂ:,(X,f)´ΓBanachL«.-k.‚5Žf
h
v
:
M
α∈E
v
X
s(α)
→X
v
,h
v
((x
s(α)
)
α∈E
v
) =
X
α∈E
v
f
α
(x
s(α)
),x
s(α)
∈X
s(α)
.
-
Y
v
= kerh
v
= {(x
s(α)
)
α∈E
v
∈
M
α∈E
v
X
s(α)
:
X
α∈E
v
f
α
(x
s(α)
) = 0}.
éu∈V,u6= v,-Y
u
= X
u
.
β∈E
v
,Kβ∈E
v
,…s(β)= r(β) = v,r(β) = s(β) 6= v.-k.‚5Žf
g
β
: Y
s(β)
= Y
v
→Y
r(β)
= Y
s(β)
= X
s(β)
,g
β
((x
s(α)
)
α∈E
v
) = x
s(β)
.
éβ∈E\E
v
,Ks(β) 6= v,r(β) 6= v,-g
β
= f
β
∈B(X
s(β)
,X
r(β)
) = B(Y
s(β)
,Y
r(β)
).
½Âσ
+
v
(Γ)BanachL«
Φ
+
Γ,v
(X,f) = (Y,g),
Ù¥Y= (Y
u
)
u∈σ
+
v
(V)
,g= (g
β
)
β∈σ
+
v
(E)
.
DOI:10.12677/pm.2022.12101941814nØêÆ
UZu§ÜH
(2)v∈V´Γu:,(X,f)´ΓBanachL«.-k.‚5Žf
b
h
v
:
M
α∈E
v
X
∗
r(α)
→X
∗
v
,
b
h
v
((x
∗
r(α)
)
α∈E
v
) =
X
α∈E
v
f
∗
α
(x
∗
r(α)
),x
∗
r(α)
∈X
∗
r(α)
.
-Y
v
= (ker
b
h
v
)
∗
,Ù¥
ker
b
h
v
= {(x
∗
r(α)
)
α∈E
v
∈
M
α∈E
v
X
∗
r(α)
:
X
α∈E
v
f
∗
α
(x
∗
r(α)
) = 0}.
éu∈V,u6= v,-Y
u
= X
∗∗
u
.
β∈E
v
,Kβ∈E
v
,…s(β)= r(β) 6= v,r(β) = s(β) = v.-k.‚5Žf
g
β
: Y
s(β)
= Y
r(β)
= X
∗∗
r(β)
→Y
r(β)
= Y
v
= (ker
b
h
v
)
∗
,
÷v:é?¿F∈X
∗∗
r(β)
,é?¿(x
∗
r(α)
)
α∈E
v
∈ker
b
h
v
,k
(g
β
F)((x
∗
r(α)
)
α∈E
v
) = F(x
∗
r(β)
).
éβ∈E\E
v
,Ks(β) 6= v,r(β) 6= v,-g
β
= f
∗∗
β
∈B(X
∗∗
s(β)
,X
∗∗
r(β)
) = B(Y
s(β)
,Y
r(β)
).
½Âσ
−
v
(Γ)BanachL«
Φ
−
Γ,v
(X,f) = (Y,g),
Ù¥Y= (Y
u
)
u∈σ
−
v
(V)
,g= (g
β
)
β∈σ
−
v
(E)
.
(3)(X,f)´ΓBanachL«.é?¿u∈V,-Y
u
=X
∗
u
.é?¿β∈
E,Kβ∈E,-g
β
=
f
∗
β
∈B(X
∗
r(β)
,X
∗
s(β)
) = B(X
∗
s(β)
,X
∗
r(β)
) = B(Y
s(β)
,Y
r(β)
).
½ÂΓBanachL«
Φ
∗
Γ
(X,f) = (Y,g),
Ù¥Y= (Y
u
)
u∈V
,g= (g
β
)
β∈E
.
e¡‰Ñd †ãΓBanachL«ƒmÓ'uÂ:vpσ
+
v
(Γ)BanachL«ƒmÓ,
'uu:vpσ
−
v
(Γ)BanachL«ƒmÓ,±9pΓBanachL«ƒmÓ.
Ún2.10Γ = (V,E,s,r)´k•†ã.
(1)v∈V´ΓÂ:,(X,f),(X
0
,f
0
)´ΓBanachL«,(Y,g)=Φ
+
Γ,v
(X,f),(Y
0
,g
0
)=
Φ
+
Γ,v
(X
0
,f
0
)X½Â2.9(1),´σ
+
v
(Γ)BanachL«.T∈Hom((X,f),(X
0
,f
0
)).-k.‚5Žf
S
v
: Y
v
→Y
0
v
,S
v
((x
s(α)
)
α∈E
v
) = (T
s(α)
(x
s(α)
))
α∈E
v
.
DOI:10.12677/pm.2022.12101941815nØêÆ
UZu§ÜH
éu∈V,u6=v,-S
u
=T
u
∈B(X
u
,X
0
u
)=B(Y
u
,Y
0
u
),KS=(S
u
)
u∈V
∈Hom((Y,g),(Y
0
,g
0
)).
PS= Φ
+
Γ,v
(T).
(2)v∈V´Γu:,(X,f),(X
0
,f
0
)´ΓBanachL«,(Y,g)=Φ
−
Γ,v
(X,f),(Y
0
,g
0
)=
Φ
−
Γ,v
(Y,f
0
)X½Â2.9(2),´σ
−
v
(Γ)BanachL«.T∈Hom((X,f),(X
0
,f
0
)).-k.‚5Žf
S
v
: Y
v
= (ker
b
h
v
)
∗
→Y
0
v
= (ker
b
h
0
v
)
∗
,
÷v:é?¿F∈Y
v
,é?¿(x
0∗
r(α)
)
α∈E
v
∈ker
b
h
0
v
,k
(S
v
F)((x
0∗
r(α)
)
α∈E
v
) = F((T
∗
r(α)
x
0∗
r(α)
)
α∈E
v
).
éu∈V,u6= v, -S
u
= T
∗∗
u
∈B(X
∗∗
u
,(X
0
u
)
∗∗
) = B(Y
u
,Y
0
u
),KS= (S
u
)
u∈V
∈Hom((Y,g),(Y
0
,g
0
)).
PS= Φ
−
Γ,v
(T).
(3)(X,f),(X
0
,f
0
)´ΓBanachL«,(Y,g)= Φ
∗
Γ
(X,f),(Y
0
,g
0
)=Φ
∗
Γ
(X
0
,f
0
)X½Â2.9(3),
´ΓBanachL«.T∈Hom((X,f),(X
0
,f
0
)).é?¿u∈V,-S
u
=T
∗
u
∈B(X
0∗
u
,X
∗
u
)
= B(Y
0
u
,Y
u
),KS= (S
u
)
u∈V
∈Hom((Y
0
,g
0
),(Y,g)).PS= Φ
∗
Γ
(T).
y²(1)†(3)y²aq[12]14Ü©¥‡¼fΦ
+
v
†C¼fΦ
∗
¥`²Œy.ey(2),
Äkaq[12]14Ü©¥‡¼fΦ
−
v
¥`²ŒyS
v
´k¿Â…S
v
∈B(Y
v
,Y
0
v
).
é?¿β∈
E
v
,Kβ∈E
v
,…s(β)=r(β)6=v,r(β)=s(β)=v.é?¿F∈X
∗∗
r(β)
,é?
¿(x
0∗
r(α)
)
α∈E
v
∈ker
b
h
0
v
,k
(S
r(β)
g
β
F)((x
0∗
r(α)
)
α∈E
v
) = (S
v
g
β
F)((x
0∗
r(α)
)
α∈E
v
) = (g
β
F)((T
∗
r(α)
x
0∗
r(α)
)
α∈E
v
) = F(T
∗
r(β)
x
0∗
r(β)
)
†
(g
0
β
S
s(β)
F)((x
0∗
r(α)
)
α∈E
v
) = (g
0
β
S
r(β)
F)((x
0∗
r(α)
)
α∈E
v
)
= (g
0
β
T
∗∗
r(
β)
F)((x
0∗
r(α)
)
α∈E
v
) = (T
∗∗
r(β)
F)x
0∗
r(β)
= F(T
∗
r(β)
x
0∗
r(β)
),
=(S
r(β)
g
β
F)((x
0∗
r(α)
)
α∈E
v
) = (g
0
β
S
s(β)
F)((x
0∗
r(α)
)
α∈E
v
),KS
r(β)
g
β
F= g
0
β
S
s(β)
F,S
r(β)
g
β
= g
0
β
S
s(β)
.
é?¿β∈E\E
v
,Ks(β) 6= v,r(β) 6= v.
S
r(β)
g
β
= T
∗∗
r(β)
f
∗∗
β
= (T
r(β)
f
β
)
∗∗
= (f
0
β
T
s(β)
)
∗∗
= (f
0
β
)
∗∗
T
∗∗
s(β)
= g
0
β
S
s(β)
.
nþŒS= (S
u
)
u∈V
∈Hom((Y,g),(Y
0
,g
0
)).
†ãσ
+
v
(Γ)UìþãpBanachL«†Ó89Ùg,ܤŒ˜‡dBRep(Γ)'
uÂ:vp‰ÆBRep(σ
+
v
(Γ)),?Œ½Â'uÂ:‡¼f.†ãσ
−
v
(Γ)Uìþãp
BanachL«†Ó89Ùg,ܤŒ˜‡dBRep(Γ)'uu:vp‰ÆBRep(σ
−
v
(Γ)),
DOI:10.12677/pm.2022.12101941816nØêÆ
UZu§ÜH
?Œ ½Â'uu:‡¼f.†ãΓUìþãpBanachL«†Ó89Ùg,ܤŒ
˜‡dBRep(Γ)p‰ÆBRep(Γ),?Œ½ÂC¼f.
½Â2.11Γ = (V,E,s,r)´k•†ã.
(1)v∈V´ΓÂ:.½Â‡¼f
Φ
+
Γ,v
: BRep(Γ) →BRep(σ
+
v
(Γ)),
Ù¥éBRep(Γ)¥é–(X,f),(Y,g) = Φ
+
Γ,v
(X,f)X½Â2.9(1),´BRep(σ
+
v
(Γ))¥é–,é
BRep(Γ)¥ÓT,S=Φ
+
Γ,v
(T)XÚn2.10(1),´BRep(σ
+
v
(Γ))¥Ó,BRep(σ
+
v
(Γ))¥Ó
ܤ´g,ܤ,XÚn2.4.
(2)v∈V´Γu:.½Â‡¼f
Φ
−
Γ,v
: BRep(Γ) →BRep(σ
−
v
(Γ)),
Ù¥éBRep(Γ)¥é–(X,f),(Y,g) = Φ
−
Γ,v
(X,f)X½Â2.9(2),´BRep(σ
−
v
(Γ))¥é–,é
BRep(Γ)¥ÓT,S=Φ
−
Γ,v
(T)XÚn2.10(2),´BRep(σ
−
v
(Γ))¥Ó,BRep(σ
−
v
(Γ))¥Ó
ܤ´g,ܤ,XÚn2.4.
(3)½ÂC¼f
Φ
∗
Γ
: BRep(Γ) →BRep(Γ),
Ù¥éBRep(Γ)¥é–(X,f),(Y,g)=Φ
∗
Γ
(X,f)X½Â2.9(3),´BRep(Γ)¥é–,éBRep
(Γ)¥ÓT,S=Φ
∗
Γ
(T)XÚn2.10(3),´BRep(Γ)¥Ó.BRep(Γ)¥Óܤ´g,Ü
¤,XÚn2.4.
e¡‰Ñ!̇½n,=`²üa‡¼fŒÏLC¼fïᘇª.
½n2.12Γ =(V,E,s,r)´k•†ã,v∈V´Γu:,Kv´ΓÂ:,σ
−
v
(Γ)=σ
+
v
(Γ),…
kXe(Ø:
(1)éΓBanachL«(X,f),k
Φ
−
Γ,v
(X,f) = Φ
∗
σ
+
v
(Γ)
(Φ
+
Γ,v
(Φ
∗
Γ
(X,f))).
(2)(X,f),(X
0
,f
0
)´ΓBanachL«,T∈Hom((X,f),(X
0
,f
0
)),K
Φ
−
Γ,v
(T) = Φ
∗
σ
+
v
(Γ)
(Φ
+
Γ,v
(Φ
∗
Γ
(T))).
y²duσ
+
v
(V) = V= σ
−
v
(V),E
v
= E
v
,K
σ
+
v
(E) = (E\E
v
)
[
E
v
= (E\E
v
)
[
E
v
= σ
−
v
(E),
DOI:10.12677/pm.2022.12101941817nØêÆ
UZu§ÜH
σ
−
v
(Γ) = σ
+
v
(Γ).
(1)P(Z,ϕ) = Φ
∗
Γ
(X,f)´ΓBanachL«,K
Z= (Z
u
)
u∈V
= (X
∗
u
)
u∈V
,ϕ= (ϕ
β
)
β∈E
= (f
∗
β
)
β∈E
.
P(W,ψ) = Φ
+
Γ,v
(Z,ϕ)´σ
+
v
(Γ)BanachL«,d½ÂŒ±†y
W
v
= ker
b
h
v
= {(x
∗
r(α)
)
α∈E
v
∈
M
α∈E
v
X
∗
r(α)
:
X
α∈E
v
f
∗
α
(x
∗
r(α)
) = 0}.
Ù¥
b
h
v
X½Â2.9(2),u∈V,u6= vž,W
u
= Z
u
= X
∗
u
.éβ∈E
v
= E
v
,k
ψ
β
((x
∗
r(α)
)
α∈E
v
) = x
∗
r(β)
,x
∗
r(α)
∈X
∗
r(α)
.
β∈E\E
v
ž,ψ
β
= ϕ
β
= f
∗
β
.P(Y,g) = Φ
∗
σ
+
v
(Γ)
(W,ψ)´σ
+
v
(Γ)BanachL«,K
Y= (Y
u
)
u∈σ
+
v
(V)
= (W
∗
u
)
u∈σ
+
v
(V)
,g= (g
β
)
β∈σ
+
v
(E)
= (ψ
∗
β
)
β∈σ
+
v
(E)
.
Y
v
= W
∗
v
= (ker
b
h
v
)
∗
,…u∈V,u6= vž,Y
u
= W
∗
u
= X
∗∗
u
.éβ∈E
v
,kβ∈E
v
,Kg
β
= ψ
∗
β
∈
B(X
∗∗
r(β)
,W
∗
v
) = B(X
∗∗
r(β)
,(ker
b
h
v
)
∗
).é?¿F∈X
∗∗
r(β)
,é?¿(x
∗
r(α)
)
α∈E
v
∈ker
b
h
v
= W
v
,k
(g
β
F)((x
∗
r(α)
)
α∈E
v
) = (ψ
∗
β
F)((x
∗
r(α)
)
α∈E
v
) = F(ψ
β
((x
∗
r(α)
)
α∈E
v
)) = F(x
∗
r(β)
).
éβ∈E\E
v
,Kβ∈E\E
v
= E\E
v
,g
β
= ψ
∗
β
= f
∗∗
β
= f
∗∗
β
.Ïd(Y,g) = Φ
−
Γ,v
(X,f),=
Φ
−
Γ,v
(X,f) = Φ
∗
σ
+
v
(Γ)
(Φ
+
Γ,v
(Φ
∗
Γ
(X,f))).
(2)PR= Φ
∗
Γ
(T),ŒR= (R
u
)
u∈V
∈Hom((Z
0
,ϕ
0
),(Z,ϕ)),…é?¿u∈V,k
R
u
= T
∗
u
: Z
0
u
= (X
0
u
)
∗
→Z
u
= X
∗
u
.
PL= Φ
+
Γ,v
(R),ŒL= (L
u
)
u∈V
∈Hom((W
0
,ψ
0
),(W,ψ)),d½ÂŒ±†y
L
v
: W
0
v
= ker
b
h
0
v
→W
v
= ker
b
h
v
,L
v
((x
0∗
r(α)
)
α∈E
v
) = (T
∗
r(α)
x
0∗
r(α)
)
α∈E
v
,
Ù¥(x
0∗
r(α)
)
α∈E
v
∈ker
b
h
0
v
.u∈V,u6= vž,k
L
u
= R
u
= T
∗
u
: W
0
u
= Z
0
u
= (X
0
u
)
∗
→W
u
= Z
u
= X
∗
u
.
DOI:10.12677/pm.2022.12101941818nØêÆ
UZu§ÜH
PS= Φ
∗
σ
+
v
(Γ)
(L),ŒS= (S
u
)
u∈V
∈Hom((Y,g),(Y
0
,g
0
)),…é?¿u∈V,k
S
u
= L
∗
u
: Y
u
= W
∗
u
→Y
0
u
= (W
0
u
)
∗
.
S
v
= L
∗
v
∈B(Y
v
,Y
0
v
) = B(W
∗
v
,(W
0
v
)
∗
) = B((ker
b
h
v
)
∗
,(ker
b
h
0
v
)
∗
).é?¿F∈Y
v
,é?¿
(x
0∗
r(α)
)
α∈E
v
∈ker
b
h
0
v
,k
(S
v
F)((x
0∗
r(α)
)
α∈E
v
) = (L
∗
v
F)((x
0∗
r(α)
)
α∈E
v
) = F(L
v
((x
0∗
r(α)
)
α∈E
v
)) = F((T
∗
r(α)
x
0∗
r(α)
)
α∈E
v
).
u∈V,u6= vž,S
u
= L
∗
u
= T
∗∗
u
.ÏdS= Φ
−
Γ,v
(T),=
Φ
−
Γ,v
(T) = Φ
∗
σ
+
v
(Γ)
(Φ
+
Γ,v
(Φ
∗
Γ
(T))).
3.‡¼fN“ê5Ÿ
½Â3.1[12]Γ=(V,E,s,r)´k•†ã,(X,f)´ΓBanachL«.v∈V´ΓÂ:.
e
X
α∈E
v
Imf
α
´X
v
4f˜m,K¡(X,f)3v´4;e
X
α∈E
v
Imf
α
= X
v
,K¡(X,f)3v´÷.
dþã½Â†ŒXe
·K3.2Γ=(V,E,s,r)´k•†ã,(X,f)´ΓBanachL«.v∈V´ΓÂ:,h
v
X½
Â2.9(1),K
(1)(X,f)3v´4⇔Imh
v
´X
v
4f˜m;
(2)(X,f)3v´÷⇔h
v
´÷.
Ún3.3Γ = (V,E,s,r)´k•†ã,(X,f)´ΓBanachL«.v∈V´ΓÂ:,K
Φ
+
Γ,v
(X,f) = 0 ⇔é?¿u∈V,u6= v,kX
u
= 0.
dž,
(1)e(X,f)„´ØŒ©),KX
v
∼
=
C;
(2)e(X,f)„3v´÷,K(X,f) = 0.
y²PΦ
+
Γ,v
(X,f) = (Y,g),Ù¥Y
v
= kerh
v
,h
v
X½Â2.9(1),…u∈V,u6= vž,Y
u
= X
u
.
eΦ
+
Γ,v
(X,f) = 0,Ké?¿u∈V,u6= v,kX
u
= Y
u
= 0.
eé?¿u∈V,u6=v,kX
u
=0,KY
u
=X
u
=0.duv´ΓÂ:,Ké?¿α∈E
v
,
s(α) 6= v,X
s(α)
= 0.ÏdY
v
= kerh
v
⊆
M
α∈E
v
X
s(α)
= 0,=Y
v
= 0.lΦ
+
Γ,v
(X,f) = (Y,g) = 0.
dž,X
u
=
n
X
v
,u= v,
0,u6= v,
u∈V,…é?¿α∈E,kf
α
= 0.
DOI:10.12677/pm.2022.12101941819nØêÆ
UZu§ÜH
(1)(X,f)„´ØŒ©),K(X,f)6=0,X
v
6= 0.edimX
v
≥2,ŒX
v
= X
1
L
X
2
,Ù
¥X
1
,X
2
´X
v
š"4f˜m.-
Y
u
=
n
X
1
,u= v,
0,u6= v,
Z
u
=
n
X
2
,u= v,
0,u6= v,
u∈V,
…é?¿α∈E,-g
α
=0,h
α
=0.-Y=(Y
u
)
u∈V
,g=(g
α
)
α∈E
,Z=(Z
u
)
u∈V
,h=(h
α
)
α∈E
,
K(Y,g),(Z,h)Ñ´Γš"BanachL«,…(X,f)=(Y,g)
L
(Z,h),ù†(X,f)´ØŒ©)gñ,
dimX
v
= 1,ÏdX
v
∼
=
C.
(2)(X,f)„3v´÷,KX
v
=
X
α∈E
v
Imf
α
= 0,X= 0,=(X,f) = 0.
Ún3.4Γ=(V,E,s,r)´k•†ã,(X,f),(Y,g)´ΓBanachL«,…(X,f)
∼
=
(Y,g).
v∈V´ΓÂ:.
(1)e(X,f)3v´4,K(Y,g)3v•´4;
(2)e(X,f)3v´÷,K(Y,g)3v•´÷.
y²du(X,f)
∼
=
(Y,g),K•3ÓT=(T
u
)
u∈V
∈Hom((X,f),(Y,g)),=é?¿u∈V,
T
u
∈B(X
u
,Y
u
)´ÓŽf,…é?¿α∈E,kT
r(α)
f
α
=g
α
T
s(α)
,Kf
α
T
−1
s(α)
=T
−1
r(α)
g
α
.é?
¿α∈E
v
,kr(α) = v,qé?¿y
s(α)
∈Y
s(α)
,-x
s(α)
= T
−1
s(α)
(y
s(α)
) ∈X
s(α)
,=y
s(α)
= T
s(α)
(x
s(α)
),
K
X
α∈E
v
g
α
(y
s(α)
) =
X
α∈E
v
g
α
T
s(α)
(x
s(α)
) =
X
α∈E
v
T
r(α)
f
α
(x
s(α)
)
=
X
α∈E
v
T
v
f
α
(x
s(α)
) = T
v
X
α∈E
v
f
α
(x
s(α)
) ∈T
v
(
X
α∈E
v
Imf
α
),

X
α∈E
v
Img
α
⊆T
v
(
X
α∈E
v
Imf
α
).ÓnŒy
X
α∈E
v
Imf
α
⊆T
−1
v
(
X
α∈E
v
Img
α
),KT
v
(
X
α∈E
v
Imf
α
) ⊆
X
α∈E
v
Img
α
,

X
α∈E
v
Img
α
= T
v
(
X
α∈E
v
Imf
α
).
(1)e(X,f)3v´4,K
X
α∈E
v
Imf
α
´X
v
4f˜m.duT
v
´ÓŽf,K
X
α∈E
v
Img
α
=
T
v
(
X
α∈E
v
Imf
α
)´Y
v
4f˜m,(Y,g)3v´4.
(2)e(X,f)3v´÷,K
X
α∈E
v
Imf
α
=X
v
.duT
v
´ÓŽf,K
X
α∈E
v
Img
α
=T
v
(
X
α∈E
v
Imf
α
)
= T
v
(X
v
) = Y
v
,(Y,g)3v´÷.
Ún3.5Γ=(V,E,s,r)´k•†ã,(X,f)´ΓBanachL«.v∈V´ΓÂ:,KN
Φ
+
Γ,v
: End(X,f) →End(Φ
+
Γ,v
(X,f))´‡“êN.
y²T∈End(X,f),S∈End(X,f),λ∈C.
DOI:10.12677/pm.2022.12101941820nØêÆ
UZu§ÜH
é?¿(x
s(α)
)
α∈E
v
∈Y
v
,k
(Φ
+
Γ,v
(T+S))
v
((x
s(α)
)
α∈E
v
) = ((T+S)
s(α)
x
s(α)
)
α∈E
v
= ((T
s(α)
+S
s(α)
)x
s(α)
)
α∈E
v
= (T
s(α)
x
s(α)
)
α∈E
v
+(S
s(α)
x
s(α)
)
α∈E
v
= (Φ
+
Γ,v
T)
v
((x
s(α)
)
α∈E
v
)+(Φ
+
Γ,v
S)
v
((x
s(α)
)
α∈E
v
)
= ((Φ
+
Γ,v
T)
v
+(Φ
+
Γ,v
S)
v
)((x
s(α)
)
α∈E
v
) = (Φ
+
Γ,v
T+Φ
+
Γ,v
S)
v
((x
s(α)
)
α∈E
v
),
(Φ
+
Γ,v
(TS))
v
((x
s(α)
)
α∈E
v
) = ((TS)
s(α)
x
s(α)
)
α∈E
v
= (T
s(α)
S
s(α)
x
s(α)
)
α∈E
v
= (Φ
+
Γ,v
T)
v
((S
s(α)
x
s(α)
)
α∈E
v
) = (Φ
+
Γ,v
T)
v
(Φ
+
Γ,v
S)
v
((x
s(α)
)
α∈E
v
) = ((Φ
+
Γ,v
T)(Φ
+
Γ,v
S))
v
((x
s(α)
)
α∈E
v
),
(Φ
+
Γ,v
(λT))
v
((x
s(α)
)
α∈E
v
) = ((λT)
s(α)
x
s(α)
)
α∈E
v
= λ(T
s(α)
x
s(α)
)
α∈E
v
= λ(Φ
+
Γ,v
T)
v
((x
s(α)
)
α∈E
v
) = (λΦ
+
Γ,v
T)
v
((x
s(α)
)
α∈E
v
),
K
(Φ
+
Γ,v
(T+S))
v
= (Φ
+
Γ,v
T+Φ
+
Γ,v
S)
v
,(Φ
+
Γ,v
(TS))
v
= ((Φ
+
Γ,v
T)(Φ
+
Γ,v
S))
v
,(Φ
+
Γ,v
(λT))
v
= (λΦ
+
Γ,v
T)
v
.
éu∈V,u6= v,k
(Φ
+
Γ,v
(T+S))
u
= (T+S)
u
= T
u
+S
u
= (Φ
+
Γ,v
T)
u
+(Φ
+
Γ,v
S)
u
= (Φ
+
Γ,v
T+Φ
+
Γ,v
S)
u
,
(Φ
+
Γ,v
(TS))
u
= (TS)
u
= T
u
S
u
= (Φ
+
Γ,v
T)
u
(Φ
+
Γ,v
S)
u
= ((Φ
+
Γ,v
T)(Φ
+
Γ,v
S))
u
,
(Φ
+
Γ,v
(λT))
u
= (λT)
u
= λT
u
= λ(Φ
+
Γ,v
T)
u
= (λΦ
+
Γ,v
T)
u
,

Φ
+
Γ,v
(T+S) = Φ
+
Γ,v
T+Φ
+
Γ,v
S,Φ
+
Γ,v
(TS) = Φ
+
Γ,v
(T)Φ
+
Γ,v
(S),Φ
+
Γ,v
(λT) = λΦ
+
Γ,v
(T).
nþŒ,Φ
+
Γ,v
: End(X,f) →End(Φ
+
Γ,v
(X,f))´‡“êN.
½n3.6Γ=(V,E,s,r)´k•†ã,v∈V´ΓÂ:,(X,f)´ΓBanachL«.e(X,f)
3v´÷,KNΦ
+
Γ,v
: End(X,f) →End(Φ
+
Γ,v
(X,f))´‡“êÓ.
DOI:10.12677/pm.2022.12101941821nØêÆ
UZu§ÜH
y²P(Y,g) = Φ
+
Γ,v
(X,f),dÚn3.5Œ•NΦ
+
Γ,v
: End(X,f) →End(Y,g)´‡“êN.
kyΦ
+
Γ,v
´ü.T∈End(X,f)÷vS=Φ
+
Γ,v
(T)=0.u∈V,u6=vž,kT
u
=S
u
=0.
duv´ΓÂ:,Ké?¿α∈E
v
,r(α)=v,s(α)6=v,T
s(α)
=0.é?¿x
s(α)
∈X
s(α)
,d
uT∈End(X,f),K
T
v
(
X
α∈E
v
f
α
(x
s(α)
)) = T
r(α)
(
X
α∈E
v
f
α
(x
s(α)
)) =
X
α∈E
v
T
r(α)
f
α
(x
s(α)
) =
X
α∈E
v
f
α
T
s(α)
(x
s(α)
) = 0.
du(X,f)3v´÷,=
X
α∈E
v
Imf
α
= X
v
,KT
v
= 0,T= (T
u
)
u∈V
= 0,¤±Φ
+
Γ,v
´ü.
eyΦ
+
Γ,v
´÷.S=(S
u
)
u∈V
∈End(Y,g).éu∈V,u6=v,-T
u
=S
u
.dué?
¿α∈E
v
,s(α) 6= v,qdu(X,f)3v´÷,=
X
α∈E
v
Imf
α
= X
v
,KŒ½ÂŽf
T
v
: X
v
→X
v
:T
v
(
X
α∈E
v
f
α
(x
s(α)
)) =
X
α∈E
v
f
α
T
s(α)
(x
s(α)
),x
s(α)
∈X
s(α)
.
e•3x
0
s(α)
∈X
s(α)
,α∈E
v
,¦
X
α∈E
v
f
α
(x
s(α)
) =
X
α∈E
v
f
α
(x
0
s(α)
),K
h
v
((x
s(α)
−x
0
s(α)
)
α∈E
v
) =
X
α∈E
v
f
α
(x
s(α)
−x
0
s(α)
) =
X
α∈E
v
f
α
(x
s(α)
)−
X
α∈E
v
f
α
(x
0
s(α)
) = 0
Ù¥h
v
X½Â2.9(1),=(x
s(α)
−x
0
s(α)
)
α∈E
v
∈kerh
v
=Y
v
.S
v
((x
s(α)
−x
0
s(α)
)
α∈E
v
)∈S
v
(Y
v
)⊆
Y
v
= kerh
v
,=
0 = h
v
(S
v
((x
s(α)
−x
0
s(α)
)
α∈E
v
)) =
X
β∈E
v
f
β
g
β
(S
v
((x
s(α)
−x
0
s(α)
)
α∈E
v
)).
é?¿β∈E
v
,Kr(β) = s(β) = v,s(β) = r(β) 6= v.duS= (S
u
)
u∈V
∈End(Y,g),K
g
β
(S
v
((x
s(α)
−x
0
s(α)
)
α∈E
v
)) = g
β
S
s(β)
((x
s(α)
−x
0
s(α)
)
α∈E
v
) = S
r(β)
g
β
((x
s(α)
−x
0
s(α)
)
α∈E
v
)
= S
s(β)
(x
s(β)
−x
0
s(β)
) = T
s(β)
(x
s(β)
−x
0
s(β)
),

0 =
X
β∈E
v
f
β
T
s(β)
(x
s(β)
−x
0
s(β)
) =
X
β∈E
v
f
β
T
s(β)
x
s(β)
−
X
β∈E
v
f
β
T
s(β)
x
0
s(β)
,
=
X
β∈E
v
f
β
T
s(β)
x
s(β)
=
X
β∈E
v
f
β
T
s(β)
x
0
s(β)
,T
v
´(½.w,T
v
´‚5.
du(X,f)3v´÷,Kh
v
´÷.dmN½n,•3~êM>0,¦é?¿(x
s(α)
)
α∈E
v
DOI:10.12677/pm.2022.12101941822nØêÆ
UZu§ÜH
∈
M
α∈E
v
X
s(α)
,
k(x
s(α)
)
α∈E
v
k≤Mkh
v
((x
s(α)
)
α∈E
v
)k= Mk
X
α∈E
v
f
α
(x
s(α)
)k.
duΓ´k•†ã,KE
v
´k•8,ŒE
v
´K8.é?¿x∈X
v
,du(X,f)3v´÷
,=
X
α∈E
v
Imf
α
=X
v
,K•3(x
s(α)
)
α∈E
v
∈
M
α∈E
v
X
s(α)
,¦x=
X
α∈E
v
f
α
(x
s(α)
).duéŽ
f(f
α
T
s(α)
)
α∈E
v
∈B(
M
α∈E
v
X
s(α)
,
M
α∈E
v
X
r(α)
),K
kT
v
xk= kT
v
(
X
α∈E
v
f
α
(x
s(α)
))k= k
X
α∈E
v
f
α
T
s(α)
(x
s(α)
)k≤
X
α∈E
v
kf
α
T
s(α)
(x
s(α)
)k
≤K
2
(
X
α∈E
v
kf
α
T
s(α)
(x
s(α)
)k
2
)
1
2
= K
2
k((f
α
T
s(α)
)
α∈E
v
)((x
s(α)
)
α∈E
v
)k
≤K
2
k(f
α
T
s(α)
)
α∈E
v
kk(x
s(α)
)
α∈E
v
k≤K
2
k(f
α
T
s(α)
)
α∈E
v
k·Mk
X
α∈E
v
f
α
(x
s(α)
)k
= K
2
Mk(f
α
T
s(α)
)
α∈E
v
kkxk,
T
v
∈B(X
v
).
é?¿α∈E
v
,Kr(α) = v.é?¿x
s(α)
∈X
s(α)
,-
x
s(γ)
=
n
x
s(α)
,γ= α,
0,γ6= α,
γ∈E
v
,
K
T
v
f
α
(x
s(α)
) = T
v
(
X
γ∈E
v
f
γ
(x
s(γ)
)) =
X
γ∈E
v
f
γ
T
s(γ)
(x
s(γ)
) = f
α
T
s(α)
(x
s(α)
),
=T
r(α)
f
α
= T
v
f
α
= f
α
T
s(α)
.é?¿α∈E\E
v
,Ks(α) 6= v,r(α) 6= v,
T
r(α)
f
α
= S
r(α)
g
α
= g
α
S
s(α)
= f
α
T
s(α)
.
ÏdT= (T
u
)
u∈V
∈End(X,f).
éu∈V,u6= v, kS
u
= T
u
= (Φ
+
Γ,v
T)
u
.duS= (S
u
)
u∈V
∈End(Y,g), Ké?¿(x
s(α)
)
α∈E
v
∈
DOI:10.12677/pm.2022.12101941823nØêÆ
UZu§ÜH
Y
v
,kS
v
((x
s(α)
)
α∈E
v
) ∈Y
v
.qdué?¿β∈E
v
,kr(β) = s(β) = v,s(β) = r(β) 6= v,K
S
v
((x
s(α)
)
α∈E
v
) = (g
β
S
v
((x
s(α)
)
α∈E
v
))
β∈E
v
= (g
β
S
s(β)
((x
s(α)
)
α∈E
v
))
β∈E
v
= (S
r(β)
g
β
((x
s(α)
)
α∈E
v
))
β∈E
v
= (S
s(β)
x
s(β)
)
β∈E
v
= (T
s(β)
x
s(β)
)
β∈E
v
= (Φ
+
Γ,v
T)
v
((x
s(α)
)
α∈E
v
),
=S
v
= (Φ
+
Γ,v
T)
v
.Φ
+
Γ,v
(T) = S.ÏdΦ
+
Γ,v
´÷.
nþ¤ã,NΦ
+
Γ,v
: End(X,f) →End(Φ
+
Γ,v
(X,f))´‡“êÓ.
Ä7‘8
I[g,䮀7(No.11971108)"
ë•©z
[1]Gabriel,P.(1972)UnzerlegbareDarstellungenI.ManuscriptaMathematica,6,71-103.
https://doi.org/10.1007/BF01298413
[2]Bernstein,I.N.,Gelfand,I.M.andPonomarev,V.A.(1973)CoxeterFunctorsandGabriel’s
Theorem.RussianMathematicalSurveys,28,17-32.
https://doi.org/10.1070/RM1973v028n02ABEH001526
[3]Brenner,S.(1967)EndomorphismAlgebrasofVectorSpaceswithDistinguishedSetsofSub-
spaces.JournalofAlgebra,6,100-114.https://doi.org/10.1016/0021-8693(67)90016-6
[4]Donovan,P.andFreislish,M.R.(1973)TheRepresentationTheoryofFiniteGraphsand
AssociatedAlgebras.In:CarletonMathematicalLectureNotes,Vol.5,CarletonUniversity,
Ottawa,1-119.
[5]Dlab,V.andRingel,C.M.(1976)IndecomposableRepresentationsofGraphsandAlgebras.
In:MemoirsoftheAMS,Vol.6,AmericanMathematicalSociety,Providence,RI.
https://doi.org/10.1090/memo/0173
[6]Gabriel, P. and Roiter,A.V. (1997)Representations of Finite-Dimensional Algebras. Springer-
Verlag,Berlin.https://doi.org/10.1007/978-3-642-58097-0
[7]Kac,V.G.(1980)InfiniteRootSystems,RepresentationsofGraphsandInvariantTheory.
InventionesMathematicae,56,57-92.https://doi.org/10.1007/BF01403155
[8]Nazarova,L.A.(1973)RepresentationofQuiversofInfiniteType.IzvestiyaAkademiiNauk
SSSR.SeriyaKhimicheskaya,37,752-791.
[9]Krause,H.andRingel,C.M.(2000)InfiniteLengthModules.Birkhauser,Basel.
https://doi.org/10.1007/978-3-0348-8426-6
DOI:10.12677/pm.2022.12101941824nØêÆ
UZu§ÜH
[10]Reiten, I.and Ringel,C.M.(2006) InfiniteDimensionalRepresentations ofCanonical Algebras.
CanadianJournalofMathematics,58,180-224.https://doi.org/10.4153/CJM-2006-008-1
[11]Kruglyak,S.A.andRoiter,A.V.(2005)LocallyScalarRepresentationsofGraphsintheCat-
egoryofHilbertSpaces.FunctionalAnalysisandItsApplications,39,91-105.
https://doi.org/10.1007/s10688-005-0022-8
[12]Enomoto, M. and Watatani, Y. (2009)Indecomposable Representations of Quivers on Infinite-
DimensionalHilbertSpaces.JournalofFunctionalAnalysis,256,959-991.
https://doi.org/10.1016/j.jfa.2008.12.011
DOI:10.12677/pm.2022.12101941825nØêÆ

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