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# Conditions for polynomials in right inverses with stationary and algebraic coefficients to be Yolterra operators

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ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1991)

Ng u y e n Va n Ma u

## to be Yolterra operators

Abstract. The conditions for polynomials in right inverses with constant coefficients were obtained by Przeworska-Rolewicz and von Trotha (see Th. 2.4.1 in [6]). In the present note we generalize this result to the case of polynomials with algebraic and stationary coefficients.

Moreover, we also give a necessary and sufficient condition for

### R1 + R2, R\Ri

to be Volterra operators, provided R 2, R 2 are Volterra right inverses of a right invertible operator D. The method used here is essentially based on the properties of generalized algebraic operators.

0

e

2

0

0

De f in it io n

0

e

d

De f in it io n

0

### P{t) =

P 0 + P i t + . . . + P Nt N,

0

(2)

De f in it io n

e

q

Th e o r e m

k = 0

e

Th e o r e m

i = 1

7 = 1

j

(3)

Theorem

[3].

u j

m

{ щ + Vf.

1,

1,

e

n ej

n

S)

7 = 0

Theorem

e

0

q

N

=1

j

i

1

0e

0

(4)

,R0 e F (X ).

Theorem 2.

n

(X)

Theorem 3.

e

a

j = о

(9)

(5)

a

s)

7=i

### This shows that I — XR is invertible for all XeC, i.e. ReV(X).

Co r o l l a r y 1.

n

1 7=1

1

Co r o l l a r y

7 = 0

e

e

d

### there exists an Aestf n S DyR such that

12 - Comment. Math. 30.2

(6)

Corollary 3.

Definition 4.

and

ker

then

is said to be an

Theorem 4.

g

11

### E

dj x Aj> dj<=jrf r \ S DtR.

j = i

j x A j

j = i

(12)

a

j = 1 J = 2

(7)

Theorem 5.

(X)

n

dim ker

0,

Definition 5.

e

(X),

Theorem6

n

(X),

n

1

Corollary 4.

п

(X),

t

Corollary 5.

F(X )

zek erZ ),

Definition

### 6. If D eR(X ) and there is an R e ^ Dn V(X), then the operators

£ a ^ ( e i A F e —î a) , s a 2 j ( @ i A F e —i A ) ,

(8)

Theorem 7.

c a

sa

casb

cbsa

### = i(eiAeiB- e - iAe - iB),

CA CB ~ S A S B = 2.(e i A e iB + e - i A e ~ i B ) -

Corollary

s

Dsa

c

Theorem 8.

e

sa

z

sa

z

z

A 2R 2)ca(z)

### = 0. which contradicts our assumption that z ф 0.

3. Operations on Volterra right inverses. Let De

e

d

i

i

1^2

Theorem 9.

e

e

e

rx

z

1

2

(16)

(9)

2

2

Th e o r e m

2

2

Ф

2

1

2e& D2

2

2

1

Ф

1

2

2

1

1

j

Ф

и Ф

2

1

1

Ф

2

1

2

1

1

1

1

2

2

z2

1

1

1

2

21

1

1

1

2

1

1

1

2

1

1

2

1

2

1

Ф 0

(10)

Th e o r e m

e

Ex a m p l e

= —

J ,

x2g[0 , 1].

XI X 2

c e

x

x x

k

Th e o r e m 11.

e

e

(X ).

e

e

e

(11)

Ex a m p l e

2

j

1

2

### is not a Volterra operator.

References

[1] N g u y e n V an M au, On algebraic properties of differential and singular integral operators with shift, Differentsial’nye Uravneniya 10 (1986), 1799-1805 (in Russian).

[2] —, Characterization o f polynomials in algebraic elements with commutative coefficients, Acta Univ. Lodz. Folia Math., to appear.

[3] —, Characterization o f polynomials in algebraic elements with constant coefficients, Demon- stratio Math. 16 (1983), 375-405.

[4] —, Arithmetical operations on algebraic operators, ibid. 22 (1989), 1109-1119.

[5] —, On solvability in closed form o f a class o f singular integral equations, Differentsial’nye Uravneniya 25 (1989), 307-311.

[6] D. P r z e w o r s k a - R o le w ic z , Algebraic Analysis, PWN-Polish Scientific Publishers and D. Reidel, Warszawa-Dordrecht 1987.

[7] —, Equations avec opérations algébriques, Studia Math. 22 (1963), 337-368.

[8] M. T a sc h e , Abstrakte lineare Differentialgleichungen mit stationàren Operator en, Math. Nachr.

78 (1977), 21-36.

INSTITUTE OF MATHEMATICS, TECHNICAL UNIVERSITY OF WARSAW PL. POLITECHNIKI 1, 00-661 WARSZAWA, POLAND

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