Îáðàçåööèòèðîâàíèÿ:ÊîðîòêîâÂ.Á.Îíåîãðàíè÷åííûõèíòåãðàëüíûõîïåðàòîðàõñêâàçèñèì- ìåòðè÷íûìèÿäðàìè//Âëàäèêàâê.ìàò.æóðí.2020.Ò.22,âûï.2.Ñ.1823.DOI:10.46698/y j

(1)

ÓÄÊ517.983

DOI10.46698/y3646-7660-8439-j

ÎÍÅÎÀÍÈ×ÅÍÍÛÕ ÈÍÒÅÀËÜÍÛÕ ÎÏÅÀÒÎÀÕ

ÑÊÂÀÇÈÑÈÌÌÅÒÈ×ÍÛÌÈ ßÄÀÌÈ

Â.Á. Êîðîòêîâ

1

Èíñòèòóòìàòåìàòèêèèì.Ñ.Ë.ÑîáîëåâàÑÎÀÍ,

îññèÿ,630090,Íîâîñèáèðñê,ïð.Àê.Êîïòþãà,4

E-mail:vitalborkorgmail.om

Àííîòàöèÿ. Â 1935 ã. îí Íåéìàí óñòàíîâèë, ÷òîïðåäåëüíûé ñïåêòð ñàìîñîïðÿæåííîãî êàðëå-

ìàíîâñêîãî èíòåãðàëüíîãî îïåðàòîðà â

L ²

^{ñîäåðæèò}

0

. Ýòîò ðåçóëüòàò áûë îáîáùåí àâòîðîì íà íåñàìîñîïðÿæåííûå îïåðàòîðû: ïðåäåëüíûé ñïåêòð îïåðàòîðà, ñîïðÿæåííîãî ê êàðëåìàíîâñêîìó

èíòåãðàëüíîìó îïåðàòîðó,ñîäåðæèò

0

. Áóäåì ãîâîðèòü,÷òîïëîòíî îïðåäåëåííûé â

L ²

^{ëèíåéíûé}

îïåðàòîð

A

óäîâëåòâîðÿåòîáîáùåííîìó óñëîâèþîí Íåéìàíà, åñëè

0

ïðèíàäëåæèò ïðåäåëüíîìó ñïåêòðó ñîïðÿæåííîãî îïåðàòîðà

A ^∗

^. ^{Îáîçíà÷èì} ^÷åðåç

B ⁰

^êëàññ ^âñåõ ^{ëèíåéíûõ} ^{îïåðàòîðîâ} ^â

L ²

^,

óäîâëåòâîðÿþùèõîáîáùåííîìóóñëîâèþîíÍåéìàíà.Àâòîðîìáûëîäîêàçàíî,÷òîêàæäûéîïðå-

äåëåííûéíà

L ²

îãðàíè÷åííûéèíòåãðàëüíûéîïåðàòîðïðèíàäëåæèòêëàññó

B ⁰

^.^{Âîçíèêàåò}^{âîïðîñ:}

âåðíî ëè àíàëîãè÷íîå óòâåðæäåíèå äëÿ ëþáîãî íåîãðàíè÷åííîãî ïëîòíî îïðåäåëåííîãî â

L ²

^èí-

òåãðàëüíîãî îïåðàòîðà? Â ñòàòüå äàåòñÿ îòðèöàòåëüíûé îòâåò íà ýòîò âîïðîñ è óñòàíàâëèâàåòñÿ

äîñòàòî÷íîåóñëîâèåïðèíàäëåæíîñòèïëîòíîîïðåäåëåííîãîâ

L ²

èíòåãðàëüíîãîîïåðàòîðàñêâàçè- ñèììåòðè÷íûìÿäðîìêëàññó

B ⁰

^.

Êëþ÷åâûåñëîâà:çàìûêàåìûéîïåðàòîð,èíòåãðàëüíûéîïåðàòîð,ÿäðîèíòåãðàëüíîãîîïåðàòîðà,

ïðåäåëüíûéñïåêòð,ëèíåéíîåèíòåãðàëüíîåóðàâíåíèå

1

-ãîèëè

2

-ãîðîäà.

Mathematial Subjet Classiation (2010):45P05,47B34.

Îáðàçåööèòèðîâàíèÿ:ÊîðîòêîâÂ.Á.Îíåîãðàíè÷åííûõèíòåãðàëüíûõîïåðàòîðàõñêâàçèñèì-

ìåòðè÷íûìèÿäðàìè//Âëàäèêàâê.ìàò.æóðí.2020.Ò.22,âûï.2.Ñ.1823.DOI:10.46698/y3646-

7660-8439-j.

Ïóñòü

(X, µ)

ïðîñòðàíñòâî ñ ïîëîæèòåëüíîé ìåðîé

µ

^,

L 0 := L 0 (X, µ)

^ñîâî-

êóïíîñòü âñåõ

µ

^{-èçìåðèìûõ}

µ

^-ïî÷òè ^âñþäó ^{êîíå÷íûõ} ^óíêöèé ^íà

X

^ñ ^{îáû÷íûì} ^îòîæ-

äåñòâëåíèåìóíêöèé,îòëè÷àþùèõñÿîäíà îòäðóãîéëèøüíàìíîæåñòâàõ

µ

^-ìåðû^íó^ëü,

L 2 := L 2 (X, µ)

ïðîñòðàíñòâî âñåõ óíêöèé èç

L 0

^ñ ñóììèðóåìûì êâàäðàòîì. ×åðåç

k · k

^è

( · , · )

^{îáîçíà÷èì}^íîðìó^è ^{ñêàëÿðíîå} ïðîèçâåäåíèåâ

L 2

^.

Ìåðà

µ

^{íàçûâàåòñÿ}

σ

^{-êîíå÷íîé,} ^åñëè ^{ñóùåñòâóþò} ^{ìíîæåñòâà}

X n ⊂ X

^,

µX n < ∞

^,

n = 1, 2, . . .

^, ^òàêèå, ^÷òî

X = S ∞

n =1 X _n

^. ^Àòîìîì ^ìåðû

µ

^{íàçûâàåòñÿ} ^{ìíîæåñòâî} ^{ïîëîæè-}

òåëüíîé ìåðû, íåïðåäñòàâèìîå â âèäå îáúåäèíåíèÿ äâóõ íåïåðåñåêàþùèõñÿ ìíîæåñòâ

ñ ïîëîæèòåëüíûìè ìåðàìè. Áóäåì ãîâîðèòü, ÷òî ìåðà

µ

^íå ^{ÿâëÿåòñÿ} ^÷èñòî ^{àòîìè÷å-}

ñêîé,åñëè â

X

^{èìååòñÿ}^{ìíîæåñòâî} ïîëîæèòåëüíîé ìåðû,íå ñîäåðæàùååàòîìîâ ìåðû

µ

^.

Âñþäó äàëåå ïðåäïîëàãàåòñÿ, ÷òî ìåðà

µ

^íå ^{ÿâëÿåòñÿ} ^÷èñòî àòîìè÷åñêîé è

σ

^{-êîíå÷íà.}

Ýòèì óñëîâèÿì óäîâëåòâîðÿåò ìåðà Ëåáåãà èçìåðèìûõ ïî Ëåáåãó ìíîæåñòâ åâêëèäîâà

ïðîñòðàíñòâà èëèâåùåñòâåííîé ÷èñëîâîé ïðÿìîé.

²⁰²⁰ ^{Êîðîòêîâ}^Â. ^Á.

(2)

Ëèíåéíûé îïåðàòîð

T : D _T ⊂ L 2 → L 0

^{íàçûâàåòñÿ} èíòåãðàëüíûì, åñëè íàéäåòñÿ îïðåäåëåííàÿíà

X × X (µ × µ)

^{-èçìåðèìàÿ}

(µ × µ)

^-ïî÷òè^âñþäó^{êîíå÷íàÿ}^óíêöèÿ

K(x, y)

òàêàÿ, ÷òî äëÿëþáîãî

f ∈ D T

T f(x) = Z

K(x, y)f (y) dµ(y) (1)

äëÿ

µ

^-ïî÷òè^âñåõ

x ∈ X

^.^{Èíòåãðàë}^â⁽¹⁾^{ïîíèìàåòñÿ}^â^{ëåáåãîâîì}^{ñìûñëå.}^{Ôóíêöèÿ}

K(x, y)

íàçûâàåòñÿÿäðîì èíòåãðàëüíîãîîïåðàòîðà

T

^. ^Áóäåì^{ãîâîðèòü,} ^÷òî^ÿäðî^{ïîðîæäàåò}^èí-

òåãðàëüíûé îïåðàòîðïî îðìóëå (1).

Îïðåäåëåíèå.Íóëüïðèíàäëåæèòïðåäåëüíîìóñïåêòðó

σ _C (H)

^{îïåðàòîðà}

H : D _H ⊂ L 2 → L 2

^,^åñëè^{ñóùåñòâóåò}îðòîíîðìèðîâàííàÿïîñëåäîâàòåëüíîñòü

{ f n } ⊂ D H

^òàêàÿ, ^÷òî

k Hf n k → 0

^ïðè

n → ∞

^.

Åñëè

T : L 2 → L 2

îãðàíè÷åííûé èíòåãðàëüíûé îïåðàòîð, òî

0 ∈ σ _C (T ^∗ )

^, ^ãäå

T ^∗

ñîïðÿæåííûé ê

T

^{îïåðàòîð} ^[1, ^ñ. ^754; ^2, ^{òåîðåìà} ^III. ^2.6℄. ^Äðóãîå äîêàçàòåëüñòâî ýòîãî ðåçóëüòàòà äàíî âêíèãå Õàëìîøà èÑàíäåðà [3, òåîðåìà 15.1℄.

Âîçíèêàåò âîïðîñ: áóäåò ëèèìåòüìåñòî âêëþ÷åíèå

0 ∈ σ _C (T ^∗ )

^, ^åñëè

T

^{ïðîèçâîëü-}

íûé íåîãðàíè÷åííûé èíòåãðàëüíûé ïëîòíî îïðåäåëåííûé çàìûêàåìûé îïåðàòîð â

L ²

^?

Îòðèöàòåëüíûé îòâåòíàýòîò âîïðîñ äàåò ñëåäóþùèé

Ïðèìåð. Ïóñòü

T 0 : L ∞ (0, 1) ⊂ L 2 (0, 1) → L 2 (0, 1)

^{ëèíåéíûé} ^{îïåðàòîð,} ^{îïðåäåëÿå-}

ìûé ðàâåíñòâîì

T 0 f =

∞

X

n =1

nw _n

1

Z

0

f χ E n

√ mE _n dy, f ∈ L ∞ (0, 1),

ãäå

{ w n }

îðòîíîðìèðîâàííûé áàçèñÓîëøà,

χ E n

õàðàêòåðèñòè÷åñêàÿ óíêöèÿìíî- æåñòâà

E n ⊂ (0, 1)

^,

{ E n }

ïîñëåäîâàòåëüíîñòü ïîïàðíî íå ïåðåñåêàþùèõñÿ ìíîæåñòâ, óäîâëåòâîðÿþùèõ óñëîâèþ

P ∞ n =1 n √

mE n < ∞

^, ^çäåñü

m

^ìåðà ^{Ëåáåãà.} ^Ò^îãäà

T 0

^çà-

ìûêàåìûéèíòåãðàëüíûé îïåðàòîðñ ÿäðîì

K 0 (x, y) =

∞

X

n =1

nw _n (x) χ _E _n (y)

√ mE _n ,

íî

0 ∈ / σ C (T ^∗ )

^.

Äåéñòâèòåëüíî, äëÿ ëþáîé óíêöèè

f

^èç

L ∞ (0, 1)

1

Z

0

T 0 f w _j dx =

1

Z

0

f j χ E _j

p mE j

dy, j = 1, 2, . . .

Ñëåäîâàòåëüíî,

T 0 ^∗

^{îïðåäåëåí} ^íà

{ w _n }

^, ^{ïîýòîìó}

T 0 ^∗

^ïëîòíî ^{îïðåäåëåí} ^è

T 0

^èìååò ^çàìû-

êàíèå îïåðàòîð

T 0 ^∗∗

^. ^Äàëåå,^äëÿ ^ëþáîé ^óíêöèè

f

^èç

L ∞ (0, 1)

^è^âñåõ

x ∈ (0, 1)

1

Z

0

| K 0 (x, y) || f (y) | dy 6

∞

X

n =1

n k f k ^∞ p

mE _n < ∞ ,

ãäå

k · k ^∞

^íîðìà ^â

L ∞ (0, 1)

^, ^òàê ^÷òî

T 0

^{çàìûêàåìûé} èíòåãðàëüíûé îïåðàòîð. Ïðè ýòîì äëÿëþáîé óíêöèè

g ∈ D _T ₀ ^∗

k T 0 ^∗ g k ² =

∞

X

n =1

n χ E n (y)

√ mE _n

1

Z

0

gw n dx

2

=

∞

X

n =1

n ²

1

Z

0

gw n dx

2

>

∞

X

n =1

1

Z

0

gw n dx

2

= k g k ² .

Ñëåäîâàòåëüíî,

0 ∈ / σ C (T ^∗ )

^.

(3)

Îáîçíà÷èì ÷åðåç

B 0

^êëàññ ^âñåõ ^{ëèíåéíûõ} ^{îïåðàòîðîâ}

H

^â

L 2

^, ^äëÿ ^{êîòîðûõ}

0 ∈ σ C (H ^∗ )

^. ^{àçëè÷íûå} ^{óñëîâèÿ} ïðèíàäëåæíîñòè îïåðàòîðîâ êëàññó

B ⁰

^äàíû ^â ^[4℄. ^Íèæå

óñòàíàâëèâàåòñÿ åùåîäíî òàêîå óñëîâèå.

Íàçîâåì ÿäðî

K(x, y)

êâàçèñèììåòðè÷íûì, åñëè

| K(x, y) | = | K(y, x) |

^äëÿ

(µ × µ)

^-ïî÷òè ^âñåõ

(x, y) ∈ X × X. (2)

Óñëîâèþ (2)óäîâëåòâîðÿþò âñå ýðìèòîâû, êîñîýðìèòîâû, ñèììåòðè÷íûå èêîñîñèì-

ìåòðè÷íûå ÿäðà.

Òåîðåìà 1. Ïóñòü

T : D T ⊂ L ² → L ²

íåîãðàíè÷åííûé ïëîòíî îïðåäåëåííûé çàìûêàåìûé èíòåãðàëüíûé îïåðàòîð ñ êâàçèñèììåòðè÷íûì ÿäðîì

K(x, y)

^. ^Åñëè ^ñóùå-

ñòâóåò âåùåñòâåííàÿ íåîòðèöàòåëüíàÿ óíêöèÿ

a ∈ L 0

^, ïîëîæèòåëüíàÿ íà ìíîæåñòâå ïîëîæèòåëüíîé ìåðû,íå ñîäåðæàùåì àòîìîâ ìåðû

µ

^, ^èóäîâëåòâîðÿþùàÿ óñëîâèþ

Z

| K(u, v) | a(v) dµ(v) ∈ L 2 ,

òî

0 ∈ σ C (T ^∗ )

^.

⊳

^{Âûáåðåì}

α > 0

^òàê, ^÷òîáû ^{ìíîæåñòâî}

E = { x ∈ X : a(x) > α }

^{ñîäåðæàëî} ^{ïîäìíî-}

æåñòâî

e

^,

0 < µe < ∞

^,^áåç ^àòîìîâ ^ìåðû

µ

^.^Ïóñòü

ϕ ∈ L 0

^è

supp ϕ := { x ∈ X : | ϕ(x) | 6 = 0 }

^.

Îáîçíà÷èì ÷åðåç

χ e

õàðàêòåðèñòè÷åñêóþ óíêöèþ ìíîæåñòâà

e

^. ^Äëÿ ^ëþáîãî

f ∈ L 2

^è

ëþáîãî

h ∈ L ∞

^ñ

supp h ⊆ e

^èìååì, ^{îáîçíà÷èâ} ^÷åðåç

k · k ^∞

^íîðìó^â

L ∞

^:

Z Z

K(x, y)f (y) dµ(y)h(x) dµ(x)

=

Z Z

K(x, y)f (y) dµ(y)χ e (x)h(x) dµ(x)

6 Z

Z

χ _e (x)K(x, y)f(y)dµ(y)

| h(x) | dµ(x) 6 Z Z

χ _e (x) | K(x, y) || f (y) | dµ(y) | h(x) | dµ(x) 6 k h k ^∞

Z Z

e

| K(x, y) | dµ(x) | f (y) | dµ(y) = k h k ^∞ Z Z

e

| K(y, x) | dµ(x) | f (y) | dµ(y)

6 1 α k h k ^∞

Z Z

e

| K (y, x) | a(x) dµ(x) | f (y) | dµ(y) 6 1

α k h k ^∞ k λ e kk f k ,

⁽³⁾

ãäå

λ _e (y) :=

Z

e

| K(y, x) | a(x) dµ(x).

Èç (3)âûòåêàåò, ÷òî äëÿëþáîãî

f ∈ D T

| (T f, h) | 6 1

α k h k ^∞ k λ e kk f k ,

ïîýòîìó

h ∈ D T ^∗

^.

Ïîëîæèì â(3)

h = χ e

^. ^Ò^îãäà^èç ⁽³⁾^{ñëåäóåò} ^äëÿ^ëþáîãî

f ∈ L 2

Z Z

χ e (x)K(x, y)f (y) dµ(y)

dµ(x) 6 1

α k λ e kk f k .

Òàêèì îáðàçîì, ÿäðî

χ _e (x)K(x, y)

^{ïîðîæäàåò} äåéñòâóþùèé èç

L 2

^â

L 1 (e, µ)

^{îãðàíè-}

÷åííûé èíòåãðàëüíûé îïåðàòîð

τ

^ñ^{íîðìîé,} ^íå ïðåâîñõîäÿùåé

1 α k λ e k

^.

(4)

Ïóñòü

{ e _m }

ïîñëåäîâàòåëüíîñòü ìíîæåñòâ èç

e

^, ^óäîâëåòâîðÿþùèõ óñëîâèþ

0 < µe m → 0

^ïðè

m → ∞

^.^{Ïîëîæèì}^â⁽³⁾

h = χ e m

^è^{îáîçíà÷èì}^÷åðåç

P F

^{îïåðàòîð}^{óìíîæå-}

íèÿíà

χ F : P F f = χ F f

^,

f ∈ L 2

^. ^Èç⁽³⁾^{ïîäîáíî} ïðåäûäóùåìóñëåäóåò, ÷òîèíòåãðàëüíûé îïåðàòîð

P _e _m τ

^ñ ^ÿäðîì

χ _e _m (x)K(x, y)

^{äåéñòâóåò} ^èç

L 2

^â

L 1 (e, µ)

^, ^{îãðàíè÷åí} ^è^åãî ^íîðìà

íåïðåâîñõîäèò

1 α k λ e m k

^, ^ãäå

λ _e _m (y) :=

Z

e m

| K(y, x) | a(x) dµ(x).

Ïóñòü

X 0 = { y ∈ X : λ e (y) < ∞ ) }

^. ^Ò^îãäà ^äëÿ ^ëþáîãî

y ∈ X 0

^è ^ëþáîãî

m λ ² _e _m (y) 6 λ ² _e (y)

^è^äëÿ^ëþáîãî

y ∈ X 0 λ ² _e _m (y) → 0

^ïðè

m → ∞

^.Ñëåäîâàòåëüíî,

k λ _e _m k ² = R

λ ² _e _m dµ → 0

ïðè

m → ∞

^è

k P e m τ k 6 _α ¹ k λ e m k → 0

^ïðè

m → ∞

^. ^Îòñþäà ^èç^[5, ^{òåîðåìà} ^I.2.9℄^{îïåðàòîð}

τ : L 2 → L 1 (e, µ)

^âïîëíå íåïðåðûâåí.

Ïóñòü

D = { f ∈ L 2 : f ∈ D _T , k f k 6 1 }

^. ^{Ìíîæåñòâî}

P _e T D = τ D

îòíîñèòåëüíî êîìïàêòíî â

L ¹ (e, µ)

^. ^{Âîçüìåì} ^{ðàâíîìåðíî} îãðàíè÷åííóþ îðòîíîðìèðîâàííóþ ñèñòåìó óíêöèé

h n

supp h n ⊆ e

^,

n = 1, 2, . . .

^Â ^{êà÷åñòâå}

{ h n }

^ìîæíî ^{âûáðàòü} îðòîíîðìèðî- âàííóþ ñèñòåìó îáîáùåííûõ óíêöèé àäåìàõåðà

r n,e

^(èõ îïðåäåëåíèå ñì., íàïðèìåð, â[5, ãë.I, 1℄). Èìååì

{ h _n } ⊂ D _T ∗

^è ^â ^ñèëó îòíîñèòåëüíîé êîìïàêòíîñòè ìíîæåñòâà

P e T D

^â

L 1 (e, µ) k T ^∗ h _n k = sup

ϕ∈D | (T ^∗ h _n , ϕ) | = sup

ϕ∈D | (h _n , T ϕ) | = sup

ϕ∈D | (χ _e h _n , T ϕ) | = sup

ϕ∈D | (h _n , χ _e T ϕ) | → 0

ïðè

n → ∞

^, ^òàê^êàê^ïî^ëåììå^èìàíà^Ëåáåãà^[3,^ñ.^125℄

| R

h n f dµ | → 0

^ïðè

n → ∞

^äëÿ

ëþáîãî

f ∈ L 1

^, ^îòêóäà

sup

f ∈ F

Z

h n f dµ

→ 0

^ïðè

n → ∞

äëÿëþáîãîîòíîñèòåëüíîêîìïàêòíîãîìíîæåñòâà

F

^â

L 1

^(è,^â^{÷àñòíîñòè,}^äëÿ

F = P _e T D

⁾

âñëåäñòâèå ðàâíîìåðíîé îãðàíè÷åííîñòè

{ h n }

^è ñóùåñòâîâàíèÿ êîíå÷íîé

ε

^-ñåòè ^äëÿ

F

äëÿëþáîãî

ε > 0

^. Ñëåäîâàòåëüíî,

0 ∈ σ C (T ^∗ )

^.

⊲

Ñëåäñòâèå. Ïóñòü

T : D T ⊂ L ² → L ²

íåîãðàíè÷åííûé ïëîòíî îïðåäåëåííûé çàìûêàåìûé èíòåãðàëüíûé îïåðàòîð ñ âåùåñòâåííûì íåîòðèöàòåëüíûì ñèììåòðè÷íûì

ÿäðîì. Åñëè â

D _T

^{ñóùåñòâóåò} âåùåñòâåííàÿ íåîòðèöàòåëüíàÿ óíêöèÿ,ïîëîæèòåëüíàÿ íàìíîæåñòâåïîëîæèòåëüíîé ìåðû, íå ñîäåðæàùåì àòîìîâ ìåðû

µ

^, ^òî

0 ∈ σ _C (T ^∗ )

^.

Çàìå÷àíèå 1. Âêëþ÷åíèå

0 ∈ σ C (T ^∗ )

^{ïîçâîëÿåò} ñóùåñòâåííî óëó÷øèòü ñâîéñòâà ÿäðà èíòåãðàëüíîãî îïåðàòîðà

T

^ñ ^{ïîìîùüþ} ^{ïåðåõîäà} ^ê ^{óíèòàðíî} ýêâèâàëåíòíîìó èí- òåãðàëüíîìó îïåðàòîðó: â [5, òåîðåìà IV. 3.7℄ äîêàçàíî, ÷òî åñëè

L ²

ñåïàðàáåëüíîå ïðîñòðàíñòâî, òî èç

0 ∈ σ C (T ^∗ )

^{ñëåäóåò,} ^÷òî ^ìîæíî ^{ïîñòðîèòü} ^{óíèòàðíûé} ^{îïåðàòîð}

U : L 2 → L 2

^òàêîé, ^÷òî

U T U ⁻¹

èíòåãðàëüíûé îïåðàòîð ñ ÿäðîì

M (x, y)

^, ^{óäîâëå-}

òâîðÿþùèìóñëîâèþÊàðëåìàíà

Z

| M(x, y) | ² dµ(y) < ∞

äëÿ

µ

x ∈ X

^è^{óñëîâèþ}^{Àõèåçåðà:}^{ñóùåñòâóåò}ïîëîæèòåëüíàÿóíêöèÿ

b ∈ L 0

òàêàÿ, ÷òî

| M (x, y) | 6 b(x)b(y)

^äëÿ

(µ × µ)

(x, y) ∈ X × X

^.

Çàìå÷àíèå 2.Ïóñòü

L 2

ñåïàðàáåëüíîåïðîñòðàíñòâî. Òîãäàèíòåãðàëüíîåóðàâíå- íèå

αz(x) − λT z(x) = f(x), f (x) ∈ L 2 ,

(5)

ãäå

T

èíòåãðàëüíûé îïåðàòîð, óäîâëåòâîðÿþùèé óñëîâèÿì òåîðåìû 1, ìîæåò áûòü ñâåäåíî ÿâíûì ëèíåéíûìíåïðåðûâíûì îáðàòèìûì ïðåîáðàçîâàíèåì ïðè

α = 0

^ê ^ýêâè-

âàëåíòíîìóèíòåãðàëüíîìóóðàâíåíèþÔðåäãîëüìà1-ãîðîäàâ

L 2

^ñ^{ÿäåðíûì}îïåðàòîðîì, à ïðè

α 6 = 0

^êýêâèâàëåíòíîìó èíòåãðàëüíîìó óðàâíåíèþ 2-ãî ðîäà â

L 2

^ñ êâàçèâûðîæ- äåííûìêàðëåìàíîâñêèì ÿäðîì

N (x, y) =

∞

X

n =1

χ g n (x)

√ µg n

f _n,λ (y), (4)

ãäå

{ g _n }

ïðîèçâîëüíàÿïîñëåäîâàòåëüíîñòüïîïàðíîíåïåðåñåêàþùèõñÿìíîæåñòâèç

X

ñ êîíå÷íûìè ïîëîæèòåëüíûìè ìåðàìè,

{ f n,λ } ⊂ L 2

^.

Ýòî óòâåðæäåíèå íåïîñðåäñòâåííî ñëåäóåò èç ïîñòðîåíèé ñòàòüè [6℄, òàê êàê â íèõ

èñïîëüçîâàëîñü ëèøü âêëþ÷åíèå

0 ∈ σ _C (T ^∗ )

^. ^{Çàìåòèì} ^åùå, ^÷òî ^â ^[7℄ ^{ïðåäëîæåíû} ^äâà

ïðèáëèæ¼ííûõìåòîäà ðåøåíèÿ èíòåãðàëüíûõ óðàâíåíèé2-ãî ðîäà â

L ²

^ñ ^ÿäðàìè^(4).

Ëèòåðàòóðà

1. ÊîðîòêîâÂ.Á.Îíåêîòîðûõñâîéñòâàõ÷àñòè÷íîèíòåãðàëüíûõîïåðàòîðîâ//Äîêë.ÀÍÑÑÑ.

1974.Ò.217,4.Ñ.752754.

2. ÊîðîòêîâÂ.Á.Èíòåãðàëüíûåîïåðàòîðû.Íîâîñèáèðñê:Èçä-âîÍîâîñèá.ãîñ.óí-òà,1977.68ñ.

3. HalmosP.R.,SunderV.S.BoundedIntegralOperatorson

L ²

Spaes.BerlinHeidelbergNewYork:

SpringerVerlag,1978.134p.

4. Êîðîòêîâ Â. Á. Îá îäíîì êëàññå ëèíåéíûõ îïåðàòîðîâ â

L ²

^/^/ ^Ñèá. ^ìàò. æóðí.2019.Ò. 60,

1.Ñ.118122.DOI:10.33048/smzh.2019.60.110 .

5. ÊîðîòêîâÂ.Á.Èíòåãðàëüíûåîïåðàòîðû.Íîâîñèáèðñê:Íàóêà,1983.224ñ.

6. Êîðîòêîâ Â. Á. Î÷àñòè÷íîêîìïàêòíûõ ïîìåðå íåîãðàíè÷åííûõëèíåéíûõ îïåðàòîðàõâ

L ²

^/^/

Âëàäèêàâê.ìàò.æóðí.2016.Ò.18,âûï.1.Ñ. 3641.DOI:10.23671/VNC.2016.1.59 45 .

7. ÊîðîòêîâÂ.Á. Èíòåãðàëüíûåóðàâíåíèÿòðåòüåãîðîäàñíåîãðàíè÷åííûìèîïåðàòîðàìè//Ñèá.

ìàò.æóðí.2017.Ò.58,2.Ñ.333343.DOI:10.17377/smzh.2017.58.2 07.

Ñòàòüÿïîñòóïèëà 22îêòÿáðÿ2019ã.

ÊîðîòêîâÂèòàëèé Áîðèñîâè÷

Èíñòèòóòìàòåìàòèêèèì.Ñ.Ë.ÑîáîëåâàÑÎÀÍ

âåäóùèéíàó÷íûéñîòðóäíèêëàáîðàòîðèèóíêöèîíàëüíîãîàíàëèçà

ÎÑÑÈß,630090,Íîâîñèáèðñê,ïð.Àê.Êîïòþãà,4

Vladikavkaz MathematialJournal

2020,Volume 22,Issue 2,P. 1823

ONUNBOUNDED INTEGRALOPERATORS

WITHQUASISYMMETRICKERNELS

Korotkov, V.B.

1

SobolevInstituteofMathematis,

4Aad.KoptyugAve.,Novosibirsk630090,Russia

Abstrat. In 1935 von Neumann established that a limit spetrum of self-adjoint Carleman integral

operatorin

L ²

^ontains

0

.This resultwasgeneralizedbythe authoronnonself-adjoint operators: the limit spetrum of the adjoint of Carleman integral operator ontains

0

. Say that a densely dened in

L ²

^linear

(6)

operator

A

^satises ^the generalized von Neumann ondition if

0

belongs to the limit spetrum of adjoint operator

A ^∗

^. ^Denote ^by

B ⁰

^the ^lass ôf âll ^linear ôperators ⁱⁿ

L ²

^satisfying ^a generalized von Neumann ondition.Theauthorprovedthat eahboundedintegraloperator,dened on

L ²

^,^belongs^to

B ⁰

^.^Thus,^the

questionarises:isananalogousassertiontrueforall unboundeddenselydenedin

L ²

întegralôperators?În

thisnote,wegiveanegativeansweronthisquestionandweestablishasuientonditionguaranteeingthat

adenselydenedin

L ²

ûnboundedîntegralôperator^withquasisymmetriliein

B ⁰

^.

Keywords:losableoperator,integraloperator,kernerofintegraloperator,limitspetrum,linearintegral

equationoftherstorseondkind.

MathematialSubjet Classiation(2010): 45P05,47B34.

For itation: Korotkov, V. B. On Unbounded Integral Operators with Quasisymmetri Kernels,

VladikavkazMath.J.,2020,vol.22,no.2,pp.1823 (inRussian).DOI:10.46698/y3646-7660-8439 -j.

Referenes

1. Korotkov, V.B. OnSome Properties ofPartially Integral Operators, Dokl. Akad. Nauk SSSR,1974,

vol.217,no.4,pp.752754(inRussian).

2. Korotkov,V. B. Integral'nye operatory [Integral Operators℄,Novosibirsk, Izd-voNovosib.Gos.Un-ta,

1977,68p.(inRussian).

3. Halmos,P. R.and Sunder, V. S.Bounded IntegralOperators on

L ²

^Spaes,^Berlin, Heidelberg, New York,SpringerVerlag,1978,134p.

4. Korotkov,V.B.OnOneClassofLinearOperatorsin

L ²

^,^SiberianMathematialJournal,2019,vol.60, no.1,pp.8992.DOI:10.1134/S003744661 901 01 05.

5. Korotkov, V. B. Integral'nye operatory [Integral Operators℄, Novosibirsk, Nauka, 1983, 224 p.

(inRussian).

6. Korotkov, V. B. On Partially Measure Compat Unbounded Linear Operators in

L ²

^, Vladikavkaz.

Math.J.,2016,vol.18,no.1,pp.3641 (inRussian).DOI:10.23671/VNC.2016.1.59 45.

7. Korotkov, V. B. Integral Equations of the Third Kind with Unbounded Operators, Siberian

MathematialJournal, 2017,vol.58,no.2,pp.255263.DOI:10.1134/S00374466170 200 70 .

ReeivedOtober22,2019

VitalyB. Korotkov

SobolevInstituteofMathematis,

4Aad.KoptyugAve.,Novosibirsk630090,Russia,

LeadingResearherofLaboratory

ofFuntionalAnalysis

Îáðàçåööèòèðîâàíèÿ:ÊîðîòêîâÂ.Á.Îíåîãðàíè÷åííûõèíòåãðàëüíûõîïåðàòîðàõñêâàçèñèì- ìåòðè÷íûìèÿäðàìè//Âëàäèêàâê.ìàò.æóðí.2020.Ò.22,âûï.2.Ñ.1823.DOI:10.46698/y j

1

1

L 2

0

0

L 2

A

0

A ∗

B 0

L 2

L 2

B 0

L 2

L 2

B 0

1

2

(X, µ)

µ

L 0 := L 0 (X, µ)

µ

µ

X

µ

L 2 := L 2 (X, µ)

L 0

k · k

( · , · )

L 2

µ

σ

X n ⊂ X

µX n < ∞

n = 1, 2, . . .

X = S ∞

n =1 X n

µ

µ

X

µ

µ

σ

T : D T ⊂ L 2 → L 0

X × X (µ × µ)

(µ × µ)

K(x, y)

f ∈ D T

T f(x) = Z

K(x, y)f (y) dµ(y) (1)

µ

x ∈ X

K(x, y)

T

σ C (H)

H : D H ⊂ L 2 → L 2

{ f n } ⊂ D H

k Hf n k → 0

n → ∞

T : L 2 → L 2

0 ∈ σ C (T ∗ )

T ∗

T

0 ∈ σ C (T ∗ )

T

L 2

T 0 : L ∞ (0, 1) ⊂ L 2 (0, 1) → L 2 (0, 1)

T 0 f =

∞

X

n =1

nw n

1

Z

0

f χ E n

√ mE n dy, f ∈ L ∞ (0, 1),

{ w n }

χ E n

L ²

L ²

A ^∗

B ⁰

L ²

L ²

B ⁰

L ²

L ²

B ⁰

n =1 X _n

T : D _T ⊂ L 2 → L 0

σ _C (H)

H : D _H ⊂ L 2 → L 2

0 ∈ σ _C (T ^∗ )

T ^∗

0 ∈ σ _C (T ^∗ )

L ²

nw _n

√ mE _n dy, f ∈ L ∞ (0, 1),

nw _n (x) χ _E _n (y)

√ mE _n ,

0 ∈ / σ C (T ^∗ )

T 0 f w _j dx =

f j χ E _j

T 0 ^∗

{ w _n }

T 0 ^∗

T 0 ^∗∗

n k f k ^∞ p

mE _n < ∞ ,

k · k ^∞

g ∈ D _T ₀ ^∗

k T 0 ^∗ g k ² =

√ mE _n

n ²