Partial integral operators of Fredholm type on Kaplansky-Hilbert module over L0

Linear equations and operators involving partial integrals appear in elasticity theory, continuum mechanics, aerodynamics and in PDE theory [1]. Self-adjoint partial integral operators arise in the theory of Schrodinger operators [2, 3]. Spectral properties of a discrete Schrodinger operator H are closely related (see [3, 4]) to the partial integral operators which participate in the presentation of operator H.

Let Q i and Q 2 be closed bounded subsets in R ^V 1 and R ^v 2 , respectively. Partial integral operator (PIO) of Fredholm type in the space L p (Qi x Q 2 ) , P ^ 1 , is an operator of the form [1]

T = T o + T i + T 2 + K,                           (1)

where operators T 0 , T 1 , T 2 and K are defined by the following formulas

T o f (^Х,У) = k o (x,y ^{) f} (x,y ⁾ ,

T i f (x,y) = I

Ω 1

k i( x,s,y ) f ( s,y ) ds,

T 2 f ^{( x,}y ) = I

Ω 2

k 2 (x,t,y)f (x,t) dt,

Kf (x,y) = J" J"

Ω 1 Ω 2

k ( x, y ; s, t ) f ( s, t ) ds dt.

Here k o , k i , k 2 and k are given measurable functions on Q i x Q ₂ , Q i x Q 2 , Q _i x Q 2 and (Q i x Q 2 ) ² , respectively, and all integrals have to be understood in the Lebesgue sense, where ds = d^ i (s) , dt = d^(t) , Ц к ( • ) — the Lebesgue measure on the ст -algebra of subsets Q k , k = 1, 2 .

Furthermore, some simple solvability conditions for the equations Tf = g were investigated by several authors (see, for example, [1] and its references). Spectral properties of the given operator has been studied in [1, 4, 5].

Nevertheless, the description of the spectra of self-adjoint PIOs with L 2 kernels remains an open question. Difficulty of this problem is connected with non-compactness of the operators T 1 and T 2 . The article studies some characteristic properties of self-adjoint partially integral operators of Fredholm type in the Kaplansky-Hilbert module L o [L 2 (Q i )] over L o (Q 2 ) . The mathematical tools from the Kaplansky–Hilbert module is used as presented in [6].

The paper is organized as follows. In Section 3 we prove the existence of an L 0 -eigenvalue for the PIO T 1 .

In Section 4 we study existence of the countable consequence of real L 0 -eigenvalues for PIO T i . In Section 5 it is given the decomposition of the PIO T i in series of V i -one-dimensional operators. In Section 5 (in section 6) is given decomposition of the PIO T 1 (the PIO T 2 ) in series of V i - ( V 2 -) one-dimensional operators.

2.    Kaplansky–Hilbert Module over L0

Recall some notions and results from the theory of Kaplansky–Hilbert modules (see [6]).

Let (Q k , X k , Ц к ) be a space with complete finite measure Ц к , L o (Q k ) -algebra of equivalence classes of all complex measurable functions on (Q k , X k , Ц к ) , where k = 1, 2 . We denote by L o L(Q i )] the set of equivalence classes of all complex measurable functions f (x, y) on Q i x Q 2 , which satisfies the condition: the integral

v ⁽ y ) = j

Ω 1

I f ^{( х,}У ^{)| 2} dm ^{( x )}

exists for almost all y E Q 2 and ^ G Lq (Q 2 ) .

We consider the map (•, ^>i : LoL(Qi)] x LoL(Qi)] ^ Lo(Q2) by rule f,g>i = j

Ω 1

^{f ( s,}y ) g ^{( s,}y ) dm i ^{( s )} .

It is clear, that the map (• , ^> i satisfies the conditions of L o (Q 2 ) -valued inner product.

For each f G L o [ L 2 (Q i )] we define L o -norm:

Il f lli <^) = V ( f.f > i (^>.

Then L o L(Q i )] is Banach-Kantorovich space over L o (Q 2 ) [6, 7]. Consequently, the space L o [L 2 (Q i )] is Kaplansky-Hilbert module over L o (Q 2 ) with the inner product (• , ^> i (ш).

If for the map A : L o L(Q i )] ^ L o [L 2 (Q i )] the equality A ( a • f + в • g) = a • Af + в • Ag is hold for all a, в € L o (Q 2 ) , f, g € L o [L 2 (Q i )] , then A is called L o (Q 2 ) -linear operator.

If for the L o (Q 2 ) -linear operator A there exists C = C (w) € L o (Q 2 ) such that, ||Af ||i (w) C C (w) | f ||i (w) for all f € L o [L 2 (Q i )] , then A is called L o (Q 2 ) -bounded operator.

For each L o (Q 2 ) -linear L o (Q 2 ) -bounded operator A we define L o (Q 2 ) -norm by the rule

| A | 1 = | A | i (w)=sup {| Af | i (w): I f | i C e } .

We say the net (£ _a ) _aG A C L o (Q 2 ) ( o' ) -converges to the element £ € L o (Q 2 ) , whenever there is a decreasing net (е в ) e e B C L o (Q 2 ) such that inf { е в : в € B } = 9 and for each в € B there is an index a(в) € A with | £ _a — £ | C е в for all a € A : а(в) C a . In this case, the element £ is called (o) -limit of the set (£ _a ) _aG A and we write £ = (o) - lim£ _a .

We know [8], that the (o) -converges of the net (£ _a ) _aG A C L o (Q 2 ) to the element £ is equivalent to converges almost everywhere to the element £ of the net (£ _a ) _aG A C Lq (Q 2 ) .

The net ( f a ) a e A in L o [L 2 (Q i )] is called (bo) -converging to f € L o [L 2 (Q i )] , if (o) - lim I f a — f ||i = 9 in L o (Q 2 ) .

Let Л2 be the Boolean algebra of idempotents in Lq(Q2). If (fa)a^A C Lo[L2(Qi)] and (na)aeA is a partition of the unit in Л2, then the series ffa na • fa (bo)-converges in Lo[L2(Qi)] and its sum is called the mixing of (fa)a^A with respect to (na)aEA. We denote this sum by mix (nafa). A subset K C Lo[L2(Qi)] is called cyclic, if mix (nafa) € K for each (fa)aGA C K and any partition of the unit (na)aGA in Л2. A subset K C Lo[L2(Qi)] is called cyclically compact, if K is cyclic and every net in K has a cyclic subset that (bo)-converges to some point of K. A subset is called relatively cyclically compact, if it is contained in a cyclically compact set.

A L o -linear operator in L o [L 2 (Q i )] is called cyclically compact, if for every L o -bounded set B in L o [L 2 (Q i )] the set A(B) is relatively cyclically compact in L o [L 2 (Q i )].

Let T i be an operator in the Kaplansky-Hilbert module L o [L 2 (Q i )] over L o (Q 2 ) given by the formula

^{( T} i ^{f )( x}, У) = j k i ⁽ x, s, y ^{)f ( s}, y) dm ^(s) .

Ω 1

Here, k i (x,s,y) is a measurable function on Q i x Q 2 .

Let the kernel k i (x, s, y) of the integral operator T i satisfy the condition

j j | k i (x,s,y) | ² d^ i (s) d^ i (x) € L o (Q 2 ).                          (4)

Ω 1 Ω 1

Then, the operator T i with values in L o (Q 2 ) is linear and bounded on L o [L 2 (Q i )]. Also, let the kernel k i (x, s,y) satisfy the condition:

k i (x, s, y) = k i (s, x, y).

Then the operator T i is a self-adjoint operator on the Kaplansky-Hilbert module L o [L 2 (Q i )] , i. e.,

( T i f,g ) i = ( f,Lg ) i .

A system { f a (x,y) } C L o [L 2 i )] is V i -orthogonal system, if ( f a , f e ) i = 9 , a = в . A V i -or-thogonal system { f _a (x, y) } C L o [L 2 (Q i )] is said to be V i -orthonormal system, if ( f _a , f _a ) i = e .

Note that, the PIO T 1 is a good example for cyclically compact operators on Kaplansky– Hilbert module [7].

3.    L0-Eigenvalue of the Partial Integral Operator T1

In this section we prove the existence of an L 0 -eigenvalue for the PIO. Put H

L 0 ^{[ L} 2 ^(Q 1 ^)] -

Theorem 3.1. The partial integral operator T 1 has non zero L 0 -eigenvalue.

<1 Put

D o = < ш E ^ 2 :

^ j | k ₁ (x,s,ш) | ² d^ 1 (x) d^ 1( s ) > 0 ► .

Ω 1 Ω 1

Then ^(Do) = 0. For each f E H, f = 9 we define subset suppQ2(f) with positive measure by the following equality suPP(f) = {ш E Q2 : (f,fЫш) = 0}-

Ω 2

Let f o E H , ||f o ^ i (^) = 0 for all ш E D o and T i f o = 9. It is clear, that Т П f o = 9 for all n E N , as: if

T k f o = 9,  T k +1 f o = 9, for some k >  1

then we get a contradiction

9 = TT+ ¹ f o ,T - ¹ fo) 1 (ш) = TTf f o ,T k fo^ 1 (ш) = 9.

We construct two sequences { f k (х,ш) } к^ ₀ , { f k (х,ш) } к^ ₀ of functions from the Kaplansky-Hilbert module L o L(Qi)] (N o = N U 0) :

f k ( x,ω )

f k ( x, ш ) =

W fk W iM’

0 ,

x E Q i , ш E supp(f k ),

Ω 2

x E Q i , ш E Q 2 \ supp(f k ),

Ω 2

f k +1( x,ш' ) = ( T i f k )(х,ш).

It follows from [9] that

I f k | 1 (ш) ^ || / к+1 Н 1 (ш), k E N,                                  (5)

and

| f k+1 | 1 (ш) • I f | 1 (ш) = ( f _fc - 1 ,f _fc +1 ) 1 (ш) = f k +i J k - iYiH , k E N.          (6)

On the other hand

|T1ffc_1|1(ш) ^ |T1|1(ш), k E N, where |T1|1(ш) E Lo(Q2) is the Lo(Q2) valued norm of the PIO Ti. Consequently,

Il f klW ) < | T 1 | 1 (ш), k E N.

Thus, for almost all ш E Q the sequence {| f k||i (w) } k e N has a finite limit А(ш) ^ 0 , i. e.,

lim ^{| f} k||1^{( ш )} = А ^{( ш )} , k ^^

for almost all ш E ^ 2 - We have А(ш) E L g (^ 2 ) , as W f k ||1 (ш) € L g (^) , k E N- From the relation (5) it follows that A = 0 . Now, we define the family of integral operators { Т 1 (ш) } on L2 (^ i ) by

T 1

^{( ш}Ы^{х )} = j Q 1

k 1 (x, s, ш ) ^ ( s ) d^ 1 (s),

у E L 2M ), ш E ^ 2 -

Then, Т1(ш) is a compact operator on L2(^i) for almost all ш E Q2- By the compactness of the operator Т1(ш) there exists subsequence fni (х,ш) such that fni+1(x,ш) = T1(ш)fni (х,ш) has a limit g(x,ш) in the Lg-norm || • W1- It is clear g E H and g = 0- Analogously, for each sequence fni+2(x, ш) = Т1(шИп+1(х, ш),   /ni+3(x, ш) = T1(ш)^ni+2(x, ш)

we obtain f n i +2 ^ h E H and f n i +3 ^ h E H by the L g -norm || • W 1 -

Using the relations (6), (7) we obtain

^Wh ^- g|l 1M = I™ ll^fn k+3 ^{- f} n k+1^W^') k ^^

= ^lim { ^{W f} n k +3 ^W 1 ^{( ш )} + ll^fn k + 1 ^W 1 ^{( ш ) —} ( ^f n k +3 , ^f n k + 1 ) 1 ^{( ш ) —} f n k +1 ^,f n k +з ) 1 ^{( ш )} } k ^^

= 0

for almost all ш E ^ 2 and so h = g. On the other hand, from the equalities

f n k +2 (x, ш) =	Г (T 1 ( ш )f nk+1 )(х, ш )	x x	E E	fi x , ш E SUpp ( f n k + 1 ) , Q 2 Q 1 , ш E Q 2 \ supp(f n k +1 ), Q 2
	^ f nk +1 H 1 (^)     , 0,
	' ( T 1 ( U ) f nk + 2 )( х,Ш ) ^ f nk +2 ^ 1 (ш)     ,	x	E	fi 1 , ш E S U pp ( f n k +2 ) ,
f n k +3 (х,ш) =				Q 2
	0,	x	E	Q 1 , ш E Q 2 \ supp(f n k +2 )
				Q 2

||Уп к +1 ^| 1 ^{( ш )} • ^f n k +2 ^{( x,ш )} = ^{( T} 1 ^{( ш ) f} n k + 1 ^{)( x,ш ) ,    ш E} ^ 2 ,

||^fn k +2||1M • ^f n k +3 ^{( х,ш )} = ^{( T} 1 ^{( ш ) f} n k +2 ^{)( x,ш ) ,    ш E} ^ 2 -

we have

It is clear that

Lim f +1 ^| 1 ^{( ш )} = ||g ^W 1 ^{( ш )} = k ^^

= ,^{lim W f} n k +3 ^| 1 ^{( ш) :} k ^^

^: ,^{lim W f} n k +2 ^W 1 ^{( ш ) :} k ^^

= W h W 1 (ш) = А(ш)-

= W h W 1 (ш)

From the equalities (8), (9) it follows that

А(ш) • h(x,ш) = Т 1 (ш)д(х,ш),

А(ш) • h(x,ш) = T 1 (ш)h(x,ш),

i. e.,

⁽ T 1 g ^{)( x},y) = ^{A (} y) • ^{h (} x,y),    ^{( T} 1 ^{h )( x,}y) = ^{A (} y) • g ⁽ x,y)-

Hence it follows that

T 1 ^{( h +} g ⁾⁽ x,y) = ^{A (} y) • ⁽ h ⁺ g ⁾⁽ x,y),   T 1 ⁽ h ^- g ⁾⁽ x,y) = ^{- A (} y) • ⁽ h ^- g ⁾⁽ x,y)-

We know, that h = 0 , g = 0- Hence we can conclude that: h + g = 0 or h — g = 0- It means that the function A(y) is an L g -eigenvalue of the PIO T 1 - >

4.    Spectral Properties of the Partial Integral Operator T1 on the Kaplansky—Hilbert Module Lo[L2(Qi)]

Theorem 4.1. For a PIO Ti the following function Ao(w) = sup^g^1=e |(Tig,g)i(w)| is nonzero and either +Ao(w) or —Ao(w) is Lo-eigenvalue of the Ti.

• <1 Put

  Q o =  w G Q 2 :

  
  

^ j | k _i (x,s,w) | ² dg _i (x) dg _i (s) > 0 > .

Ω 1 Ω 1

From the T i = 9 it follows that A o (w) = 0 for all w G Q o , i. e., A o = 9. It is clear, that there is a sequence of V i -normal functions { g _n } n=i , in which a limit exists

(o) - lim ( T i g n ,g n ) i (w) = A(w), n ^^

and A(w) is a real function on Q ₂ , where A(w) = +A o (w) or — A o (w). Consequently, A o G L o (Q 2 ) and supp(A) = Q o -

By cyclical compactness of the PIO T i there exists a subsequence { g n i } i=i with

(bo) - lim ( T i g n k )(x,y) = h(x,y). k ^^

Clearly, supp Q 2 (h) = Q q . From the equality

IIT i g n k ^{— A} • g n k 111 = IIT i g n k 111

^{2 A} • ^{( T} 1 ^g n k ^,g n k ) i + ^A 2

we obtain

^{( o ) - lim} IITi g n k ^{— A} • g n k 111 = h h h i ^{— A} 2 . k ^^

However,

IIT i g n k ||i ^{( w )} ^ ^A o ^{( w )} • ||g n k ||i ^{( w )} = A ⁽ w 1 .

Therefore,

| h | i (w) < | A(w) | .

From this and (11) we have | h | i (w) = | A(w) | . Thus,

^{( o ) - lim} IITi g n k ^{— A} • g n k IIi = ⁹. k ^^

Hence, it follows that

Tifo = A • fo, where

f o (x,w) = <

h(x,ω) λ(ω) ^,

0,

x

x

Put

n o (w) =

1,

0,

G Q i , w G

G Q i , w G

^SU PP ^{( A )} ,

Ω 2

Q 2 \ supp(A). >

Ω 2

w G supp(A),

Ω 2

w G Q 2 \ supp(A).

Ω 2

Remark 4.1. Every element Z G L o (Q 2 ) , n o Z = A is L o -eigenvalue of the PIO T i .

Theorem 4.2. The PIO Ti has a finite or countable sequence of Vi-orthonormal eigenfunctions

Ф 1 ( х,у ) ,Ф2 ( х,у ) , ... ,Ф п (x,y),...

corresponding to a system of real nonzero Lo-eigenvalues

A i (w), A2 ( w ) , ..., А п (ш),...,

where

^{| А} 1 ^{( ш )|} ^ ^{| А} 2 ⁽ Ш) | ^ . . . ^ ^{| А} п ⁽ Ш) | ^ . . .

Moreover, for each f (x, y) € Lq[L2(^i)] the equality

^

» f n iM-w-E i f^

к=1

holds.

• <1 Put H i = H and T ¹ = T i . By the Theorem 4.1 there is such element ф 1( х,у ) € H i that Т ⁽¹'⁾ Ф 1 = A i • Ф1, where A i is a real function on Q 2 and А 1 (ш) = ± sup ^ _g ^ ₁ = _e |( Tig, g ) i (w) | . We define the Kaplansky-Hilbert submodule H 2 = H i ©1 { Ф 1 } . It is clear that if f € H 2 , then T ⁽²⁾ f € H 2 from the equality ( f, ф 1 ) 1 = 6 it follows that

(T 1⁽¹⁾ f,Ф 1 > ₁ = ( f,T^ i ) i = ( f,A 1 • Ф 1 ) 1 = 6.

We define an operator T₂ ²⁾ on the H 2 by

T f = T i⁽¹⁾ f, f € H 2 .

The operator Ti"2) is a selfadjoint PIO on the H2. If Ti"2) = 6, then we apply Theorem 4.1 to the operator T22) and find an element ф2(х, y) € H2 such that T22^2 = А2 • Ф2, where А2 is a real function on Q2 and А2(ш) = ± sup    |^T1(2)g,g^1 (w) |. As ф2(х,у) € H2, ЦФ2Н1 = e, geH2, hgbi=e we have (ф2,фi)i = 6. Therefore,

^{| А}2М | =     sup     ^|(7 i⁽¹⁾ g^,g^ ₁ (w)¹ <  sup     ^|(T i⁽¹⁾ g^,g> ₁ (w) = ^{| А} 1 (w) | .

g E H 2 , h g b i = e                         g & H i , l|g | i = e

Continuing this process we obtain a sequence of Kaplansky-Hilbert submodules H k+1 = H k ©1 { ф к } , where ф к € H k are eigenfunctions of the PIO T i with T^ k = А к • ф к .

If T n) is a zero operator for some n € N then we obtain the finite system V ₁ -orthonormal eigenfunctions ф 1 (х, y), ф 2 (х, y),... , ф _п- 1 (х, y) corresponding to the system of nonzero Lq - eigenvalues А 1 (ш), А ₂ (ш),..., А _п- 1 (ш), such that

|А1(ш)| ^ |А2(Ш)| ^ . .. ^ |Ап i(w)| and

^{| А} к ^{( w )|} =     sup       TiEg^g _i ⁽ wi.

g ^ H k , | g | i = e

If T ₁ ^{( n )} = 6 for each n € N then we obtain an infinite system V 1 -orthonormal eigenfunctions { Ф к } fc =i corresponding to the system of L o —eigenvalues А к = 6 . However, the equality

T1(w)фk(x,w) = Ак(ш) • фк(x,w),    k € N, is correct for almost all ш E Q2. It follows that limk^^ Ак(ш) = 0 for almost all ш E ^2, because Т1(ш) is a compact operator for almost all ш E ^2-

Let / = T i h , h E H and g = h — ^2m₌i( h, ф к > 1 • ф к - Here m is the number of eigenfunctions of the system { ф к } when the system { ф к } is a finite set, and m is equal to arbitrary natural number otherwise. By the equality

(д,фк>1 = 9, k E {1, 2,... ,m} we have g E Hm+i. Consequently, we have

»Т1д»1(ш) < |T1m+1,|2(^) -И(ш), i. e.,

m

Th -У/Мк >1 • Тфк к=1

(ш) <  ^ Т 1^{, т +1}^ | ² (ш) J| g l ? (ш).

We have { h,ф k >1 • Т\ф к = ( Tlh,ф k >i • ф к and | g | 1 ^ | h | 1 - Hence by the inequality (13)

we obtain

m f — ^(f, фк>1 • фк к=1

(ш) « Ц т '"*¹¹ | 2 (ш) •| h | 2 (ш).

If the number of elements of the system { ф к } is equal to m then т ( ^{т +1)} = 9 and we have

m f = ^(f, фк>1 • фк-к=

If the sequence { ф к } is infinite then from the inequality (14) it follows that

m f — ^(f, фк>1 • фк к=

(ш) ^ A mm+1 (ш) •«(ш),

i. e.,

m

9 < I f || 2 (ш) — £ |( /,ф к > 1 (ш) | ² <  Am, ₊1 (ш) • | h | 2 (ш). к=

Thus as m ^ to , we get

∞

II/^{| 2 ( ш )} = ^{( o ) -} ^ К/^,ф к Ы^{ш )} 1 ² - ▻ к=1

5.    Decomposition of the Partial Integral Operator T1 in Series of V1-One-Dimensional Operators

Definition 5.1. If for an operator A : H ^ H there are V 1 -orthonormal functions { ф к } n=1 C H and some system of functions { д к } П=1 C H , such that

n a/ = ^(/-дк>1 • фк, / e h к=1

then the operator A is called the V 1 - n - dimensional operator, here H = Lq [ L2 (^ 1 )]-

Theorem 5.1. For the PIO T i there is a system of V i-orthonormal functions { ф к (x,y) } and a sequence of real Lq (Q 2 ) -eigenvalues Х к (w) such that for all h G L o [L 2 (Q i )] the following conditions hold:

1 ° . h = h o + ( bo )- E^x t h, ф к ) i • ф к , h o G Ker ( T i) .

2 ° . T i h = ( bo )- H =i X k •( h,ф k ) i • ф к .

3 ° . 1 Х к ( ш )1 > | Х к+1 (ш) | , k G N.

4 ° . (o)-lim k ^^ X k = 0.

• <1 By Theorem 4.2 there are a system of V y -orthonormal functions { ф к (x,y) } and a sequence of L o -eigenvalues Х к (w) such that Tф k = Х к • ф к and for each f = Th we get the equality

^ ^f = ^{( bo ) -} ^ ( f, ^ф к ) 1 • ^ф к , k =1

where ( f, ф к ) i = Х к • ( h, ф к ) i .

Thus, for all h G H

^

T i h = ( bo )-^ Х к •( h,ф k > i • ф к .

k =1

If we denote h o = h — (bo) - Efc =_i ( h, ф к ) i • ф к , then

^

h = h o + (bo) - ^(h,ф k > i • ф к , T i h o = 0.

k =1

The properties 3 ° and 4 ° follows from the Theorem 4.2. Theorem 5.1 can also be proven by using Theorem 3.5 in the article of A. G. Kusraev [10]. >

Theorem 5.2. For all positive functions e(w) G L o (Q 2 ), ^(Q 2 \ supp(e)) = 0 there exist a Vi-finite dimensional operator T y on the Kaplansky-Hilbert module L o [L 2 (Q i )], such that ll^T i ^— TliM < Е ^{( ш )} .

• < By the Theorem 5.1 there is a system of Vi-orthonormal functions {фк(x,y)} and a sequence of Lo-eigenvalues Хк (w) for which the properties 1°-4° hold. We define the Vi-finite dimensional operator T^e:

6.    Decomposition of the Partial Integral Operator T2 in Series of V2-One-Dimensional Operators

n

Tih = У^ Хк • (h, фк)1 • фк• k=1

It follows that iTih — Tieh|2(ш) ^ ХП+1(ш) • {|(h, фк+1)1(ш)|2 + |(h, фк+2)1(ш)|2 + ...} ^ Xk(ш)|h|2(ш).

Hence, for | Х _п +1 (ш) | <  e ( w ) we have ||T _i — T^l^w ) <  е(ш). >

We denote by L o [L 2 (^ 2 )] the set of equivalence classes of all complex measurable functions f (x,y) on Q i x Q 2 , which satisfied the condition: the integral

W^{x )} = I ^{| f ( x,}y ^{)| 2 d}^ 2 ⁽y )

Ω 2

exist for almost all x G Q i and ф G L o (Q i ).

We define L o (Q i ) -valued inner product on L o [L 2 (Q 2 )] by

(М2 = /

Ω 1

f (x,t)g(x,t) d^ 2 (t).

For each f € L o [L 2 (Q 2 )] we define L o -norm: ||f Ц 2 (u) = 1/ ( f, f ^(u). Then L o L(^ 2 )] is a Banach-Kantorovich space over L o (Q i ) - Consequently, the space L o [L 2 (Q 2 )] is a Kaplansky-Hilbert module over L o (Q i ) with the inner product (• , -^ (u).

Let T2 be an operator in the Kaplansky-Hilbert module Lo[L2(Q2)] over Lo(Qi) given by the formula

⁽ T 2 ^{f )( х,}У)

j k 2( x,t,y ) f (x,t) d№(t).

Ω 2

Here, k 2 (x,t,y) is measurable function on Q i x Q 2 .

Assume that the kernel k2(x,s,y) of the integral operator T2 satisfies the condition j j |k2(x,t,y)|2 d^(t) d^2(y) € Lo(Qi).

Ω 2 Ω 2

Then, the operator T 2 is linear and L o (Q i ) -bounded operator on L o [L 2 (Q 2 )] - If the kernel k 2 (x, s, y ) satisfy of the condition k 2 (x, t, y ) = k 2 (x, y, t), then the operator T 2 is a self-adjoint operator on the Kaplansky-Hilbert module L o [L 2 (Q 2 )], i- e.,

Ш2 = ( f,T 2 g > 2 -

A system { f _a (x, y) } € L o [L 2 (Q 2 )] is said V 2 -orthogonal system, if ( f _a , f p ) 2 = 9, a = в - A V 2 -orthogonal system { f _a (x,y) } C L o [L 2 (Q 2 )] is said V 2 -orthonormal system, if ( f _a ,f O )2 = e

Note that, the PIO T 2 is cyclically compact on the Kaplansky–Hilbert module

^L o ^{[ L} 2 ^(Q 2 ^{)] [7]}-

Theorem 6.1. For the PIO T2 there is a system of V2-orthonormal functions {фk(x,y)} and a sequence of real Lo(Qi)-eigenvalues Zk (u) such that, for all h € Lo[L2 (Q2)] the following hold:

• 1 ° . h = h o + (bo)-E fc =_i ( h,Ф k ) i • Ф к , h o € Ker ( T 2 );

• 2 ° . T 2 h = (bo)- ^ k=_i Z k • ( h, Ф к ) i • Ф к , where

• 3 ° . | Z k (u) | > | Z k+i (u) | , k € N;

• 4 ° . (o)-lim k -x Z k = 9.