CUBO A Mathematical Journal
Vol.16, No¯ 01, (73–80). March 2014

On certain functional equation in semiprime rings and
standard operator algebras

Nejc Širovnik
1

Department of Mathematics and Computer Science,

Faculty of Natural Sciences and Mathematics,

University of Maribor,

Koroš ka cesta 160, 2000 Maribor, Slovenia

nejc.sirovnik@uni-mb.si

ABSTRACT

The main purpose of this paper is to prove the following result, which is related to

a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be

the algebra of all bounded linear operators on X and let A(X) ⊆ L(X) be a standard

operator algebra. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the

relation 2D(An) = D(An−1)A+An−1D(A)+D(A)An−1+AD(An−1) for all A ∈ A(X),

where n ≥ 2 is some fixed integer. In this case D is of the form D(A) = [A, B] for all

A ∈ A(X) and some fixed B ∈ L(X), which means that D is a linear derivation. In

particular, D is continuous.

RESUMEN

El propósito principal de este art́ıculo es probar el siguiente resultado, el cual se rela-

ciona a un resultado clásico de Chernoff. Sea X un espacio de Banach real o complejo,

sea L(X) el álgebra de todos los operadores lineales acotados en X y sea A(X) ⊆ L(X)

una álgebra de operadores estándar. Supongamos que existe una aplicación lineal

D : A(X) → L(X) satisfaciendo la relación 2D(An) = D(An−1)A + An−1D(A) +

D(A)An−1 + AD(An−1) para todo A ∈ A(X), donde n ≥ 2 es algún entero fijo. En

este caso D es de la forma D(A) = [A, B] para todo A ∈ A(X) y algún B ∈ L(X) fijo,

lo que significa que D es una derivación lineal. En particular, D es continua.

Keywords and Phrases: Prime ring, semiprime ring, Banach space, standard operator algebra,

derivation, Jordan derivation.

2010 AMS Mathematics Subject Classification: 16N60, 46B99, 39B42

1This research has been supported by the Research Council of Slovenia.



74 Nejc Širovnik CUBO
16, 1 (2014)

This research has been motivated by the work of Vukman [19]. Throughout, R will represent

an associative ring with center Z(R). As usual we write [x, y] for xy − yx. Given an integer n ≥ 2,

a ring R is said to be n-torsion free if for x ∈ R, nx = 0 implies x = 0. Recall that a ring R is prime

if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0)

implies a = 0.

Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear

mapping D : B → A is called a linear derivation in case D(xy) = D(x)y + xD(y) holds for all

pairs x, y ∈ B. In case we have a ring R, an additive mapping D : R → R is called a derivation

if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case

D(x2) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists such

a ∈ R that D(x) = [x, a] holds for all x ∈ R.

Every derivation is a Jordan derivation. The converse is in general not true. A classical result

of Herstein [9] asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation. A

brief proof of Herstein theorem can be found in [2]. Cusack [7] generalized Herstein theorem to

2-torsion free semiprime rings (see [3] for an alternative proof). Herstein theorem has been fairly

generalized by Beidar, Brešar, Chebotar and Martindale [1]. For results concerning derivations in

rings and algebras we refer to [5, 11, 16, 17, 18, 19], where further references can be found. Let

X be a real or complex Banach space and let L(X) and F(X) denote the algebra of all bounded

linear operators on X and the ideal of all finite rank operators in L(X), respectively. An algebra

A(X) ⊆ L(X) is said to be standard in case F(X) ⊂ A(X). Let us point out that any standard

operator algebra is prime.

Motivated by the work of Brešar [4], Vukman [19] has recently conjectured that in case we

have an additive mapping D : R → R, where R is a 2-torsion free semiprime ring satisfying the

relation

2D(xyx) = D(xy)x + xyD(x) + D(x)yx + xD(yx) (1)

for all pairs x, y ∈ R, then D is a derivation. Note that in case a ring has the identity element,

the proof of Vukman’s conjecture is immediate. Namely, in this case the substitution y = e in the

relation (1), where e stands for the identity element, gives that D is a Jordan derivation and then it

follows from Cusack’s generalization of Herstein theorem that D is a derivation. The substitution

y = xn−2 in the relation (1) gives

2D(xn) = D(xn−1)x + xn−1D(x) + D(x)xn−1 + xD(xn−1),

which leads to the following conjecture.

Conjecture 0.1. Let R be a semiprime ring with suitable torsion restrictions and let D : R → R

be an additive mapping. Suppose that

2D(xn) = D(xn−1)x + xn−1D(x) + D(x)xn−1 + xD(xn−1)

holds for all x ∈ R and some fixed integer n ≥ 2. In this case D is a derivation.



CUBO
16, 1 (2014)

On certain functional equation in semiprime rings . . . 75

It is our aim in this paper to prove the conjecture above in case a ring has the identity element.

Theorem 0.2. Let n ≥ 2 be some fixed integer, let R be a n!-torsion free semiprime ring with the

identity element and let D : R → R be an additive mapping satisfying the relation

2D(xn) = D(xn−1)x + xn−1D(x) + D(x)xn−1 + xD(xn−1)

for all x ∈ R. In this case D is a derivation.

Proof. We have the relation

2D(xn) = D(xn−1)x + xn−1D(x) + D(x)xn−1 + xD(xn−1) (2)

and let us denote the identity element of R by e. Putting e for x in the above relation, we obtain

D(e) = 0. (3)

Let y be any element of the center Z(R). Putting x + y in the above relation, we obtain

2

n∑

i=0

(

n

i

)

D(xn−iyi) =

( n−1∑

i=0

(

n−1

i

)

D(xn−1−iyi)

)

(x + y)

+

( n−1∑

i=0

(

n−1

i

)

xn−1−iyi
)

D(x + y)

+ D(x + y)

( n−1∑

i=0

(

n−1

i

)

xn−1−iyi
)

+ (x + y)

( n−1∑

i=0

(

n−1

i

)

D(xn−1−iyi)

)

.

Using (2) in the above relation and rearranging it in sense of collecting together terms involving

equal number of factors of y, we obtain

n−1∑

i=1

fi(x, y) = 0,

where fi(x, y) stands for the expression of terms involving i factors of y. Replacing x by x + 2y,

x + 3y, . . . , x + (n − 1)y in turn in the relation (2) and expressing the resulting system of n − 1

homogeneous equations of variables fi(x, y), i = 1, 2, . . . , n − 1, we see that the coefficient matrix

of the system is a Vandermonde matrix















1 1 . . . 1

2 22 . . . 2n−1

...
...

...
...

n − 1 (n − 1)2 . . . (n − 1)n−1















.



76 Nejc Širovnik CUBO
16, 1 (2014)

Since the determinant of this matrix is different from zero, it follows that the system has only a

trivial solution. In particular,

fn−2(x, e) = 2
(

n

n−2

)

D(x2) −
(

n−1

n−2

)

D(x)x −
(

n−1

n−3

)

D(x2)

−
(

n−1

n−2

)

xD(x) −
(

n−1

n−3

)

x2a −
(

n−1

n−2

)

D(x)x

−
(

n−1

n−3

)

ax2 −
(

n−1

n−2

)

xD(x) −
(

n−1

n−3

)

D(x2),

where a denotes T(e). After some calculation and considering the relation (3), we obtain

(n(n − 1) − (n − 1)(n − 2))D(x2) = 2(n − 1)(D(x)x + xD(x)).

Since R is 2(n − 1)-torsion free, the above relation reduces to

D(x2) = D(x)x + xD(x)

for all x ∈ R. In other words, D is a Jordan derivation and Cusack’s generalization of Herstein

theorem now implies that D is a derivation, which completes the proof.

In the proof of Theorem 0.2 we used methods similar to those used by Vukman and Kosi-Ulbl

in [10]. We proceed with the following result in the spirit of Conjecture 0.1.

Theorem 0.3. Let X be a real or complex Banach space and let A(X) be a standard operator

algebra on X. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation

2D(An) = D(An−1)A + An−1D(A) + D(A)An−1 + AD(An−1)

for all A ∈ A(X) and some fixed integer n ≥ 2. In this case D is of the form D(A) = [A, B] for all

A ∈ A(X) and some fixed B ∈ L(X), which means that D is a linear derivation.

In case n = 3 the above relation reduces to Theorem 4 in [19]. Let us point out that in Theorem

0.3 we obtain as a result the continuity of D under purely algebraic assumptions concerning D,

which means that Theorem 0.3 might be of some interest from the automatic continuity point of

view. For results concerning automatic continuity we refer the reader to [8] and [13]. In the proof

of Theorem 0.3 we use Herstein theorem, the result below and methods that are similar to those

used by Kosi-Ulbl and Vukman in [12].

Theorem 0.4. Let X be a real or complex Banach space, let A(X) be a standard operator algebra

on X and let D : A(X) → L(X) be a linear derivation. In this case D is of the form D(A) = [A, B]

for all A ∈ A(X) and some fixed B ∈ L(X).

Theorem 0.4 has been proved by Chernoff [6] (see also [14, 15]).

Proof of the Theorem 0.3. We have the relation

2D(An) = D(An−1)A + An−1D(A) + D(A)An−1 + AD(An−1) (4)



CUBO
16, 1 (2014)

On certain functional equation in semiprime rings . . . 77

for all A ∈ A(X). Let us first restrict our attention on F(X). Let A be from F(X) and let P ∈ F(X)

be a projection with AP = PA = A. Putting P for A in the relation (4), we obtain

D(P) = D(P)P + PD(P). (5)

Putting A + P for A in the relation (4), we obtain, similary as in the proof of Theorem 0.2, the

relation

2

n∑

i=0

(

n

i

)

D(An−iPi) =

( n−1∑

i=0

(

n−1

i

)

D(An−1−iPi)

)

(A + P)

+

( n−1∑

i=0

(

n−1

i

)

An−1−iPi
)

D(A + P)

+ D(A + P)

( n−1∑

i=0

(

n−1

i

)

An−1−iPi
)

+ (A + P)

( n−1∑

i=0

(

n−1

i

)

D(An−1−iPi)

)

.

Using (4) and (5) in the above relation and rearranging it in sense of collecting together terms

involving equal number of factors of P, we obtain

n−1∑

i=1

fi(A, P) = 0,

where fi(A, P) stands for the expression of terms involving i factors of P. Replacing A by A + 2P,

A + 3P, . . . , A + (n − 1)P in turn in the relation (4) and expressing the resulting system of n − 1

homogeneous equations of variables fi(A, P), i = 1, 2, . . . , n − 1, we see that the coefficient matrix

of the system is a Vandermonde matrix















1 1 . . . 1

2 22 . . . 2n−1

...
...

...
...

n − 1 (n − 1)2 . . . (n − 1)n−1















.

Since the determinant of this matrix is different from zero, it follows that the system has only a

trivial solution. In particular,

fn−1(A, P) = 2
(

n

n−1

)

D(A) −
(

n−1

n−1

)

D(P)A −
(

n−1

n−2

)

D(A)P

−
(

n−1

n−1

)

PD(A) −
(

n−1

n−2

)

AD(P) −
(

n−1

n−1

)

D(A)P

−
(

n−1

n−2

)

D(P)A −
(

n−1

n−1

)

AD(P) −
(

n−1

n−2

)

PD(A).

The above relation reduces to

2D(A) = D(A)P + AD(P) + D(P)A + PD(A) (6)



78 Nejc Širovnik CUBO
16, 1 (2014)

and putting A2 for A in the above relation, we obtain

2D(A2) = D(A2)P + A2D(P) + D(P)A2 + PD(A2). (7)

As the previously mentioned system of n − 1 homogeneous equations has only a trivial solution,

we also obtain

fn−2(A, P) = 2
(

n

n−2

)

D(A2) −
(

n−1

n−2

)

D(A)A −
(

n−1

n−3

)

D(A2)P

−
(

n−1

n−2

)

AD(A) −
(

n−1

n−3

)

A2D(P) −
(

n−1

n−2

)

D(A)A

−
(

n−1

n−3

)

D(P)A2 −
(

n−1

n−2

)

AD(A) −
(

n−1

n−3

)

PD(A2).

The above relation now reduces to

n(n − 1)D(A2) = 2(n − 1)(D(A)A + AD(A)) +

+
(

n−1

n−3

)

(D(A2)P + A2D(P) + D(P)A2 + PD(A2)).

Applying the relation (7) in the above relation, we obtain

n(n − 1)D(A2) = 2(n − 1)(D(A)A + AD(A)) + (n − 1)(n − 2)D(A2),

which reduces to

D(A2) = D(A)A + AD(A). (8)

From the relation (6) one can conclude that D maps F(X) into itself. We therefore have a linear

mapping D, which maps F(X) into itself and satisfies the relation (8) for all A ∈ F(X). In other

words, D is a Jordan derivation on F(X) and since F(X) is prime, it follows, according to Herstein

theorem, that D is a derivation on F(X). Applying Theorem 0.4 one can conclude that D is of the

form

D(A) = [A, B] (9)

for all A ∈ F(X) and some fixed B ∈ L(X). It remains to prove that (9) holds for all A ∈ A(X)

as well. For this purpose we introduce D1 : A(X) → L(X) by D1(A) = [A, B] and consider the

mapping D0 = D−D1. The mapping D0 is obviously linear, satisfies the relation (4) and vanishes

on F(X). It is our aim to prove that D0 vanishes on A(X) as well. Let A ∈ A(X), let P be a

one-dimensional projection and let us introduce S ∈ A(X) by S = A + PAP − (AP + PA). We have

SP = PS = 0. Obviously, D0(S) = D0(A). By the relation (4) we now have

D0(S
n−1)S + Sn−1D0(S) + D0(S)S

n−1 + SD0(S
n−1)

= 2D0(S
n) = 2D0(S

n + P) = 2D0((S + P)
n)

= D0((S + P)
n−1)(S + P) + (S + P)n−1D0(S + P)

+ D0(S + P)(S + P)
n−1 + (S + P)D0((S + P)

n−1)

= D0(S
n−1

)S + D0(S
n−1

)P + Sn−1D0(S) + PD0(S)

+ D0(S)S
n−1

+ D0(S)P + SD0(S
n−1

) + PD0(S
n−1

).



CUBO
16, 1 (2014)

On certain functional equation in semiprime rings . . . 79

From the above relation it follows that

D0(S
n−1)P + PD0(S) + D0(S)P + PD0(S

n−1) = 0.

Since D0(S) = D0(A), we can rewrite the above relation as

D0(A
n−1

)P + PD0(A) + D0(A)P + PD0(A
n−1

) = 0. (10)

Putting 2A for A in the above relation, we obtain

2n−1D0(A
n−1)P + 2PD0(A) + 2D0(A)P + 2

n−1PD0(A
n−1) = 0. (11)

In case n = 2, the relation (10) implies that

PD0(A) + D0(A)P = 0. (12)

In case n > 2, the relations (10) and (11) give the above relation (12). Multiplying the above

relation from both sides by P, we obtain

PD0(A)P = 0.

Right multiplication by P in the relation (12) gives PD0(A)P + D0(A)P = 0, which is reduced by

the above relation to

D0(A)P = 0.

Since P is an arbitrary one-dimensional projection, it follows from the above relation that D0(A) =

0 for all A ∈ A(X), which completes the proof of the theorem.

Received: January 2013. Accepted: February 2014.

References

[1] K. I. Beidar, M. Brešar, M. A. Chebotar, W. S. Martindale 3rd: On Herstein’s Lie map

Conjectures II, J. Algebra 238 (2001), 239-264.

[2] M. Brešar, J. Vukman: Jordan derivations on prime rings, Bull. Austral. Math. Soc. Vol. 37

(1988), 321-322.

[3] M. Brešar: Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988),

1003-1006.

[4] M. Brešar: Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218-228.

[5] M. Brešar, J. Vukman: Jordan (θ,φ)-derivations, Glasnik Mat. 16 (1991), 13-17.



80 Nejc Širovnik CUBO
16, 1 (2014)

[6] P. R. Chernoff: Representations, automorphisms and derivations of some Operator Algebras,

J. Funct. Anal. 2 (1973), 275-289.

[7] J. Cusack: Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324.

[8] H. G. Dales: Automatic continuity, Bull. London Math. Soc. 10 (1978), 129-183.

[9] I. N. Herstein: Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-

1119.

[10] I. Kosi-Ulbl, J. Vukman: A note on derivations in semiprime rings, Int. J. Math. & Math.

Sci., 20 (2005), 3347-3350.

[11] I. Kosi-Ulbl, J. Vukman: On derivations in rings with involution, Internat. Math. J. Vol. 6

(2005), 81-91.

[12] I. Kosi-Ulbl, J. Vukman: An identity related to derivations of standard operator algebras and

semisimple H∗-algebras, CUBO A Mathematical Journal, 12 (2010), 95-103.

[13] A. M. Sinclair: Automatic continuity of linear operators, London Math. Soc. Lecture Note

Ser. 21, Cambridge University Press, Cambridge, London, New York and Melbourne (1976).

[14] P. Šemrl: Ring derivations on standard operator algebras, J. Funct. Anal. Vol. 112 (1993),

318-324.

[15] J. Vukman: On automorphisms and derivations of operator algebras, Glasnik Mat. Vol. 19

(1984), 135-138.

[16] J. Vukman: On derivations of algebras with involution, Acta Math. Hungar. 112 (3) (2006),

181-186.

[17] J. Vukman: On derivations of standard operator algebras and semisimple H∗-algebras, Studia

Sci. Math. Hungar. 44 (2007), 57-63.

[18] J. Vukman: Identities related to derivations and centralizers on standard operator algebras,

Taiwan. J. Math. Vol. 11 (2007), 255-265.

[19] J. Vukman: Some remarks on derivations in semiprime rings and standard operator algebras,

Glasnik. Mat. Vol. 46 (2011), 43-48.