SUPERSTABILITY FOR GENERALIZED MODULE LEFT DERIVATIONS AND GENERALIZED MODULE DERIVATIONS ON A BANACH MODULE (II)
HUAI-XIN CAO, JI-RONG LV, AND J. M. RASSIAS COLLEGE OFMATHEMATICS ANDINFORMATIONSCIENCE
SHAANXINORMALUNIVERSITY
XI’AN710062, P. R. CHINA
[email protected] [email protected] PEDAGOGICALDEPARTMENT
SECTION OFMATHEMATICS ANDINFORMATICS
NATIONAL ANDCAPODISTRIANUNIVERSITY OFATHENS
ATHENS15342, GREECE
Received 12 January, 2009; accepted 12 May, 2009 Communicated by S.S. Dragomir
ABSTRACT. In this paper, we introduce and discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module.
Key words and phrases: Superstability, Generalized module left derivation, Generalized module derivation, Module left derivation, Module derivation, Banach module.
2000 Mathematics Subject Classification. Primary 39B52; Secondary 39B82.
1. INTRODUCTION
The study of stability problems was formulated by Ulam in [28] during a talk in 1940: “Under what conditions does there exist a homomorphism near an approximate homomorphism?” In the following year 1941, Hyers in [12] answered the question of Ulam for Banach spaces, which states that ifε > 0andf : X → Y is a map with a normed spaceX and a Banach spaceY such that
(1.1) kf(x+y)−f(x)−f(y)k ≤ε,
for allx, yinX, then there exists a unique additive mappingT :X →Y such that
(1.2) kf(x)−T(x)k ≤ε,
for allx inX. In addition, if the mapping t 7→ f(tx)is continuous in t ∈ Rfor each fixedx in X, then the mapping T is real linear. This stability phenomenon is called the Hyers-Ulam stability of the additive functional equation f(x+y) = f(x) +f(y). A generalized version
This subject is supported by the NNSFs of China (No. 10571113, 10871224).
013-09
of the theorem of Hyers for approximately additive mappings was given by Aoki in [1] and for approximate linear mappings was presented by Th. M. Rassias in [26] by considering the case when the left hand side of the inequality (1.1) is controlled by a sum of powers of norms [25]. The stability of approximate ring homomorphisms and additive mappings were discussed in [6, 7, 8, 10, 11, 13, 14, 21].
The stability result concerning derivations between operator algebras was first obtained by P. Semrl in [27]. Badora [5] and Moslehian [17, 18] discussed the Hyers-Ulam stability and the superstability of derivations. C. Baak and M. S. Moslehian [4] discussed the stability of J∗-homomorphisms. Miura et al. proved the Hyers-Ulam-Rassias stability and Bourgin-type superstability of derivations on Banach algebras in [16]. Various stability results on derivations and left derivations can be found in [3, 19, 20, 2, 9]. More results on stability and superstability of homomorphisms, special functionals and equations can be found in J. M. Rassias’ papers [22, 23, 24].
Recently, S.-Y. Kang and I.-S. Chang in [15] discussed the superstability of generalized left derivations and generalized derivations. In the present paper, we will discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module.
To give our results, let us give some notations. LetA be an algebra over the real or complex fieldFandX be anA-bimodule.
Definition 1.1. A mappingd:A →A is said to be module-X additive if (1.3) xd(a+b) =xd(a) +xd(b) (a, b∈A, x∈X).
A module-X additive mapping d : A → A is said to be a module-X left derivation (resp., module-Xderivation) if the functional equation
(1.4) xd(ab) =axd(b) +bxd(a) (a, b∈A, x∈X) (resp.,
(1.5) xd(ab) = axd(b) +d(a)xb (a, b∈A, x∈X)) holds.
Definition 1.2. A mappingf :X →Xis said to be module-A additive if (1.6) af(x1+x2) = af(x1) +af(x2) (x1, x2 ∈X, a∈A).
A module-A additive mapping f : X → X is called a generalized module-A left deriva- tion (resp., generalized module-A derivation) if there exists a module-Xleft derivation (resp., module-Xderivation)δ:A →A such that
(1.7) af(bx) =abf(x) +axδ(b) (x∈X, a, b∈A) (resp.,
(1.8) af(bx) =abf(x) +aδ(b)x (x∈X, a, b ∈A)).
In addition, if the mappingsf andδ are all linear, then the mappingf is called a linear gener- alized module-A left derivation (resp., linear generalized module-A derivation).
Remark 1. LetA =XandA be one of the following cases:
(a) a unital algebra;
(b) a Banach algebra with an approximate unit.
Then module-A left derivations, module-A derivations, generalized module-A left derivations and generalized module-A derivations onA become left derivations, derivations, generalized left derivations and generalized derivations onA as discussed in [15].
2. MAINRESULTS
Theorem 2.1. Let A be a Banach algebra, X a Banach A-bimodule, k and l be integers greater than1, andϕ :X×X×A ×X →[0,∞)satisfy the following conditions:
(a) lim
n→∞k−n[ϕ(knx, kny,0,0) +ϕ(0,0, knz, w)] = 0 (x, y, w ∈X, z ∈A).
(b) lim
n→∞k−2nϕ(0,0, knz, knw) = 0 (z ∈A, w ∈X).
(c) ϕ(x) :=˜ P∞
n=0k−n+1ϕ(knx,0,0,0)<∞(x∈X).
Suppose that f : X → X and g : A → A are mappings such that f(0) = 0, δ(z) :=
n→∞lim
1
kng(knz)exists for allz ∈A and
(2.1)
∆1f,g(x, y, z, w)
≤ϕ(x, y, z, w) for allx, y, w ∈Xandz∈A where
∆1f,g(x, y, z, w) = fx k +y
l +zw
+fx k − y
l +zw
− 2f(x)
k −2zf(w)−2wg(z).
Thenf is a generalized module-A left derivation andgis a module-Xleft derivation.
Proof. By takingw=z = 0, we see from (2.1) that
(2.2)
fx k + y
l
+fx k −y
l
− 2f(x) k
≤ϕ(x, y,0,0)
for allx, y ∈X. Lettingy= 0and replacingxbykxin (2.2), we get (2.3)
f(x)− f(kx) k
≤ 1
2ϕ(kx,0,0,0) for allx∈X. Hence, for allx∈X, we have from (2.3) that
f(x)− f(k2x) k2
≤
f(x)− f(kx) k
+
f(kx)
k −f(k2x) k2
≤ 1
2ϕ(kx,0,0,0) + 1
2k−1ϕ(k2x,0,0,0).
By induction, one can check that (2.4)
f(x)− f(knx) kn
≤ 1 2
n
X
j=1
k−j+1ϕ(kjx,0,0,0)
for allx inX andn = 1,2, . . . . Letx ∈ X andn > m. Then by (2.4) and condition (c), we obtain that
f(knx)
kn −f(kmx) km
= 1 km
f(kn−m·kmx)
kn−m −f(kmx)
≤ 1 km · 1
2
n−m
X
j=1
k−j+1ϕ(kj·kmx,0,0,0)
≤ 1 2
∞
X
s=m
k−s+1ϕ(ksx,0,0,0)
→0 (m→ ∞).
This shows that the sequencen
f(knx) kn
o
is a Cauchy sequence in the BanachA-moduleXand therefore converges for allx∈ X. Putd(x) = lim
n→∞
f(knx)
kn for everyx∈Xandf(0) =d(0) = 0. By (2.4), we get
(2.5) kf(x)−d(x)k ≤ 1
2ϕ(x)˜ (x∈X).
Next, we show that the mappingdis additive. To do this, let us replacex, ybyknx, knyin (2.2), respectively. Then
1 knf
knx
k +kny l
+ 1
knf knx
k −kny l
− 1
k · 2f(knx) kn
≤k−nϕ(knx, kny,0,0)
for allx, y ∈X. If we letn→ ∞in the above inequality, then the condition (a) yields that
(2.6) dx
k + y l
+dx k − y
l
= 2 kd(x)
for allx, y ∈ X. Sinced(0) = 0, takingy = 0 andy = klx, respectively, we see thatd xk
=
d(x)
k andd(2x) = 2d(x)for allx∈X, and then we obtain thatd(x+y) +d(x−y) = 2d(x)for allx, y ∈X. Now, for allu, v ∈X, putx= k2(u+v), y = 2l(u−v). Then by (2.6), we get that
d(u) +d(v) =d x
k +y l
+d
x k − y
l
= 2
kd(x) = 2 kd
k
2(u+v)
=d(u+v).
This shows thatdis additive.
Now, we are going to prove thatf is a generalized module-A left derivation. Letting x = y= 0in (2.1), we get
kf(zw) +f(zw)−2zf(w)−2wg(z)k ≤ϕ(0,0, z, w), that is
(2.7) kf(zw)−zf(w)−wg(z)k ≤ 1
2ϕ(0,0, z, w)
for all z ∈ A andw ∈ X. By replacingz, w with knz, knwin (2.7) respectively, we deduce that
(2.8)
1
k2nf k2nzw
−z 1
knf(knw)−w 1
kng(knz)
≤ 1
2k−2nϕ(0,0, knz, knw) for allz ∈A andw∈X. Lettingn → ∞, condition (b) yields that
(2.9) d(zw) =zd(w) +wδ(z)
for all z ∈ A and w ∈ X. Since d is additive, δ is module-X additive. Put ∆(z, w) = f(zw)−zf(w)−wg(z). Then by (2.7) we see from condition (a) that
k−nk∆(knz, w)k ≤ 1
2k−nϕ(0,0, knz, w)→0 (n→ ∞) for allz ∈A andw∈X. Hence
d(zw) = lim
n→∞
f(knz·w) kn
= lim
n→∞
knzf(w) +wg(knz) + ∆(knz, w) kn
=zf(w) +wδ(z)
for allz∈A andw∈X. It follows from (2.9) thatzf(w) =zd(w)for allz ∈A andw∈X, and thend(w) = f(w)for allw ∈ X. Sincedis additive,f is module-A additive. So, for all a, b∈A andx∈Xby (2.9),
af(bx) =ad(bx) =abf(x) +axδ(b) and
xδ(ab) = d(abx)−abf(x)
=af(bx) +bxδ(a)−abf(x)
=a(d(bx)−bf(x)) +bxδ(a)
=axδ(b) +bxδ(a).
This shows that ifδis a module-Xleft derivation onA, thenf is a generalized module-A left derivation onX.
Lastly, we prove thatg is a module-X left derivation onA. To do this, we compute from (2.7) that
f(knzw)
kn −zf(knw)
kn −wg(z)
≤ 1
2k−nϕ(0,0, z, knw) for allz ∈A and allw∈X. By lettingn → ∞, we get from condition (a) that
d(zw) = zd(w) +wg(z)
for all z ∈ A and allw ∈ X. Now, (2.9) implies that wg(z) = wδ(z) for allz ∈ A and all w∈X. Hence,gis a module-Xleft derivation onA. This completes the proof.
Corollary 2.2. LetA be a Banach algebra,X a BanachA-bimodule,ε ≥0,p, q, s, t∈[0,1) andkandlbe integers greater than1. Suppose thatf :X →Xandg :A →A are mappings such thatf(0) = 0,δ(z) := lim
n→∞
1
kng(knz)exists for allz ∈A and
(2.10)
∆1f,g(x, y, z, w)
≤ε(kxkp+kykq+kzkskwkt)
for allx, y, w ∈Xand allz ∈A (00 := 1). Thenf is a generalized module-A left derivation andg is a module-Xleft derivation.
Proof. It is easy to check that the function
ϕ(x, y, z, w) = ε(kxkp+kykq+kzkskwkt)
satisfies conditions (a), (b) and (c) of Theorem 2.1.
Corollary 2.3. LetA be a Banach algebra with unite,ε ≥ 0, andk andlbe integers greater than1. Suppose thatf, g:A →A are mappings withf(0) = 0such that
∆1f,g(x, y, z, w) ≤ε
for allx, y, w, z ∈A. Thenf is a generalized left derivation andg is a left derivation.
Proof. By taking w = e in (2.8), we see that the limit δ(z) := lim
n→∞
1
kng(knz) exists for all z ∈A. It follows from Corollary 2.2 and Remark 1 thatfis a generalized left derivation andg
is a left derivation. This completes the proof.
Lemma 2.4. LetX, Y be complex vector spaces. Then a mappingf :X →Y is linear if and only if
f(αx+βy) = αf(x) +βf(y) for allx, y ∈Xand allα, β ∈T:={z ∈C:|z|= 1}.
Proof. It suffices to prove the sufficiency. Suppose thatf(αx+βy) = αf(x) +βf(y)for all x, y ∈ X and allα, β ∈ T :={z ∈ C : |z| = 1}. Thenf is additive andf(αx) = αf(x)for allx∈Xand allα∈T.Letαbe any nonzero complex number. Take a positive integernsuch that|α/n|<2. Take a real numberθsuch that0≤a:=e−iθα/n < 2. Putβ = arccosa2. Then α=n(ei(β+θ)+e−i(β−θ))and therefore
f(αx) = nf(ei(β+θ)x) +nf(e−i(β−θ)x)
=nei(β+θ)f(x) +ne−i(β−θ)f(x) =αf(x)
for allx∈X. This shows thatf is linear. The proof is completed.
Theorem 2.5. Let A be a Banach algebra, X a Banach A-bimodule, k and l be integers greater than1, andϕ :X×X×A ×X →[0,∞)satisfy the following conditions:
(a) lim
n→∞k−n[ϕ(knx, kny,0,0) +ϕ(0,0, knz, w)] = 0 (x, y, w ∈X, z ∈A).
(b) lim
n→∞k−2nϕ(0,0, knz, knw) = 0 (z ∈A, w ∈X).
(c) ϕ(x) :=˜ P∞
n=0k−n+1ϕ(knx,0,0,0)<∞ (x∈X).
Suppose that f : X → X and g : A → A are mappings such that f(0) = 0, δ(z) :=
n→∞lim
1
kng(knz)exists for allz ∈A and
(2.11)
∆3f,g(x, y, z, w, α, β)
≤ϕ(x, y, z, w)
for allx, y, w ∈X,z ∈A and allα, β ∈T:={z ∈C:|z|= 1}, where∆3f,g(x, y, z, w, α, β) stands for
f αx
k +βy l +zw
+f
αx k − βy
l +zw
− 2αf(x)
k −2zf(w)−2wg(z).
Thenfis a linear generalized module-A left derivation andgis a linear module-Xleft deriva- tion.
Proof. Clearly, the inequality (2.1) is satisfied. Hence, Theorem 2.1 and its proof show thatf is a generalized left derivation andg is a left derivation onA with
(2.12) f(x) = lim
n→∞
f(knx)
kn , g(x) = f(x)−xf(e) for everyx∈X. Takingz =w= 0in (2.11) yields that
(2.13)
f
αx k + βy
l
+f αx
k − βy l
−2αf(x) k
≤ϕ(x, y,0,0)
for allx, y ∈Xand allα, β ∈T. If we replacexandywithknxandknyin (2.13) respectively, then we see that
1 knf
αknx
k + βkny l
+ 1
knf
αknx
k − βkny l
− 1 kn
2αf(knx) k
≤k−nϕ(knx, kny,0,0)
→0
asn → ∞for allx, y ∈Xand allα, β ∈T. Hence,
(2.14) f
αx k +βy
l
+f αx
k −βy l
= 2αf(x) k
for allx, y ∈Xand allα, β ∈T. Sincef is additive, takingy= 0in (2.14) implies that
(2.15) f(αx) =αf(x)
for allx∈ Xand all α∈T. Lemma 2.4 yields thatf is linear and so isg. Next, similar to the proof of Theorem 2.3 in [15], one can show thatg(A)⊂Z(A)∩rad(A). This completes the
proof.
Corollary 2.6. LetA be a complex semi-prime Banach algebra with unite, ε ≥0,p, q, s, t ∈ [0,1)andk andl be integers greater than 1. Suppose thatf, g : A → A are mappings with f(0) = 0and satisfy following inequality:
(2.16)
∆3f,g(x, y, z, w, α, β)
≤ε(kxkp+kykq+kzkskwkt)
for allx, y, z, w ∈A and allα, β ∈T(00 := 1). Thenf is a linear generalized left derivation andg is a linear left derivation which mapsA into the intersection of the center Z(A)and the Jacobson radical rad(A)ofA.
Proof. SinceA has a unite, lettingw =ein (2.8) shows that the limitδ(z) := lim
n→∞
1
kng(knz) exists for allz ∈A. Thus, using Theorem 2.5 forϕ(x, y, z, w) =ε(kxkp+kykq+kzkskwkt) yields thatf is a linear generalized left derivation andg is a linear left derivation sinceA has a unit. Similar to the proof of Theorem 2.3 in [15], one can check that the mappingg mapsA into the intersection of the center Z(A)and the Jacobson radical rad(A)ofA. This completes
the proof.
Corollary 2.7. LetA be a complex semiprime Banach algebra with unite, ε ≥ 0, k andlbe integers greater than1. Suppose thatf, g : A → A are mappings withf(0) = 0and satisfy the following inequality:
∆3f,g(x, y, z, w, α, β) ≤ε
for allx, y, z, w ∈A and allα, β ∈T. Thenf is a linear generalized left derivation andgis a linear left derivation which mapsA into the intersection of the center Z(A)and the Jacobson radical rad(A)ofA.
Remark 2. Inequalities (2.10) and (2.16) are controlled by their right-hand sides by the “mixed sum-product of powers of norms", introduced by J. M. Rassias (in 2007) and applied afterwards by K. Ravi et al. (2007-2008). Moreover, it is easy to check that the function
ϕ(x, y, z, w) =Pkxkp+Qkykq+Skzks+Tkwkt
satisfies conditions (a), (b) and (c) of Theorem 2.1 and Theorem 2.5, whereP, Q, T, S ∈[0,∞) andp, q, s, t ∈[0,1)are all constants.
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