H.Reisi

Department of Mathematics,Semnan University,Semnan,Iran.

A Perturbation of Jensen ?-Derivations from K(H) into K(H)

H.Reisi?

Department of Mathematics,Semnan University,Semnan,Iran.

.Let’s take H as an inf i nite–dimensional Hilbert space and K(H)be the set of all compact operators on H.Using Spectral theorem for compact self–adjoint operators,we prove the Hyers–Ulam stability of Jensen ?-derivations from K(H)into K(H).

Jensen ?-derivation,C? -algebra,Hyers–Ulam stability.

AMS SubjectClassif i cations:52B10,65D18,68U05,68U07

1 Introduction

In a Hilbert space H,an operator T in B(H)is called a compact operator if the image of unit ball of H under T is a compact subset of H.Note that if the operator T:H ?→ H is compact,then the adjoint of T is compact,too.The set of all compact operators on H is shown by K(H).It is easy to see that K(H)is a C?-algebra[1].Moreover,every operator on H with f i nite range is compact.The set of all f i nite range projections on Hilbert space H is denoted by P(H).

An approximate unit for a C?-algebra A is an increasing net(uλ)λ∈Λof positive elements in the closed unit ball of A such that a=limλauλ=limλuλa for all a ∈ A.Every C?-algebra admits an approximate unit[2].

Example 1.1.Let H be a Hilbert space with orthonormal basisThe C?- algebra K(H)is non–unital since dim(H)= ∞ .If Pnis a projection on Ce1+···+Cen,then the increasing sequenceis an approximate unit for K(H).

Theorem 1.1(see[2]).Let T:H ? → H be a compact self–adjoint operator on Hilbert space H. Then there is an orthonormal basis of H consisting of eigenvectors of T.The nonzero eigenvalues of T are from fi nite or countably in fi nite setof real numbers and T=where Pkis the orthogonal projection on the fi nite–dimensional space of eigenvectors corresponding to eigenvalues.If the number of nonzero eigenvalues is countably in fi nite,then the series converges to T in the operator norm.

Theproblemofstability offunctionalequationsoriginatedfromaquestionofUlam[5] concerning the stability of group homomorphisms:let(G1,?)be a group and let(G2,?,d) be a metric group with the metric d(·,·).Given ε＞ 0,does there exist a δ(ε)＞ 0 such that if a mapping h:G1?→ G2satisf i es the inequality

for all x,y ∈ G1,then there exists a homomorphism H:G1→ G2with

for all x ∈ G1?If the answer is aff i rmative,we would say that the equation of homomorphism H(x?y)=H(x)?H(y)is stable.Thus,the stability question of functional equations is that how the solutions of the inequality differ from those of the given functional equation.

Hyers[3]gave the f i rst aff i rmative answer to the question of Ulam for Banach spaces. Let X and Y be Banach spaces.Assume that f:X ?→ Y satisf i es

for all x,y ∈ X and some ε＞ 0.Then,there exists a unique additive mapping T:X ?→ Y such that

for all x ∈ X.Also,if the function tf(tx)from R to Y is continuous for each f i xed x ∈ X, then T is an R-linear function.This method is called the direct method or Hyers–Ulam stability of functional equations.

Note that if f is continuous,then the function rf(rx)from R into Y is continuous for all x ∈ X.Therefore T is R-linear.

Def i nition1.1.Let X andY bereallinear spaces.Forn∈{2,3,4,···}themapping f:X?→Y is called a Jensen mapping of n-variable,if f for each x1,···,xn∈ X satisf i es the following equation

In2003,J.M.Rassiasand M.J.Rassias[4]investigatedtheUlamstability ofJensenand Jensen type mappings by applying the Hyers method.In 2012,M.Eshaghi Gordji and S.Abbaszadeh[6]investigated the Hyers–Ulam stability of Jensen type and generalized n-variable Jensen type functional equations in fuzzy Banach spaces.

Def i nition 1.2.Let A be a C?-algebra.A mapping d:A ?→ A with d(a?)=d(a)?for all a ∈ A(?-preserving property)is called a Jensen ?-derivation if d satisf i es

and

for all λ ∈ C,x1,···,xn∈ A and n ∈ {2,3,4,···}.

Def i nition 1.3.LetA be a C?-algebra.A mapping d:A?→A with d(a?)=d(a)?for all a∈A (?-preserving property)and d(0)=0 is called a Jensen Jordan ?-derivation if d satisf i es

and

for all λ ∈ C,x1,···,xn∈ A and n ∈ {2,3,4,···}.

B.E.Johnson[7]investigated almost algebra ?-homomorphism between Banach ?-algebras.Recently,M.Eshaghi Gordji et al.have investigated several stability results on homomorphisms and Jordan homomorphisms on C?-algebras(see[8]).

In the present paper,using spectral theorem for compact self–adjoint operators,we prove that every almost-Jensen ?-preserving map ?:K(H)?→ K(H)satisfying ?(T2nP)= T2n?(P)+ ?(T)2nP for all P∈ P(H),can be Jensen ?-derivation.Also,we showthat every almost-Jensen ?-preserving map ? :K(H)?→ K(H)satisfying

for all P ∈ P(H),can be Jensen Jordan ?-derivation.

2 Jensen ?-derivations and Jensen Jordan ?-derivations

From now on,we suppose that H is an inf i nite dimensional Hilbert space,K(H)is the set of all compact operators and P(H)is the set of all f i nite range projections on H.

It is easy to see that if a Jensen mapping ? satisf i es the condition ?(0)=0,then ? is additive.We use this fact in the main results of this paper.

Lemma 2.1.Assume that X and Y be linear spaces.If a mapping f:X ? → Y is additive and for each fi xed x ∈ X ,f(λx)= λ f (x)for all λ ∈:={:0 ≤ θ ＜ θ0≤ 2 π}.Then f is C-linear. Proof.If λ belongs to T1,then there exists θ ∈ [0,2π]such that λ=It follows from→ 0 as n → ∞ t hat there exists n0∈ N such that λ1=belongs toand f(λx)=f== λ f (x)for all x ∈ X .Let t∈ ( 0,1).Putting t1=t+i(1 ? t2),t2=t? i( 1 ? t2),then we have t=and t1,t2∈ T1.It follows that

If λ ∈ B1:={λ ∈ C;|λ|≤ 1},then there exists θ∈ [0,2π]such that λ =|λ|eiθ.It follows that

for all x ∈ X.If λ ∈ C then there exist n0∈ N(from→ 0 as → ∞)such that λ0=∈ B1and for all x ∈ X

Thus,we complete the proof.

Theorem 2.1.Let H be an inf i nite dimensional Hilbert space and ε＞ 0 is given;if a continuous mapping ? :K(H)?→ K(H)with ?(0)=0 satisf i es the following conditions:

Then there exists a unique Jensen ? -derivation D:K(H)?→ K (H)such that

for all T ∈ K(H).

Proof.From condition(2),there exists a unique Jensen mapping D:K(H)?→ K (H)with D(T)=such that

for all T ∈ K(H)(see[4,6]).Note that D(0)=0,thus D is additive.Now,it follows from condition(2)and Lemma 2.1 that D is C-linear.

It follows from condition(1)that

for all T ∈ K(H)and P ∈ P(H).So,since D is linear we get

for all T ∈ K(H)and P ∈ P(H).By tending n to inf i nity in the last equality above,we obtain

for all T ∈ K(H)and P ∈ P(H).By(2.1)and(2.2),we have ?(T)P=D(T)P for all T ∈ K(H) and P ∈ P(H).

Now,we show that D ≡ ?.Let{Pm} ? P(H)be an approximate unit of K(H),then we get

for all T ∈ K(H).

Given S,T ∈ K(H),there are compact self adjoint operators S1and S2such that S= S1+iS2.According to Theorem 1.1 we have

where Pk∈P(H)and αk,βk∈C forall k∈{1,2,3,···}.Itfollows fromlinearity andcontinuity of D and T that

The last equality obtained by continuity of ?.Indeed,uniformly.Hence

From the condition(3)we conclude that

Hence,D is ?-preserving.This means that D is a Jensen ?-derivation.

Corollary 2.1.Let H be an inf i nite dimensional Hilbert space and ε＞ 0 is given;if a mapping ? :K(H)?→ K(H)with ?(0)=0 satisf i es the following conditions:

Then ? is Jensen Jordan ?-derivation.

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Received 6 January 2014;Accepted(in revised version)28 July 2016

?Corresponding author.Email address:hamidreza.reisi@gmail.com(H.Reisi)