Some functional equations on standard operator algebras
Tóm tắt
The main purpose of this paper is to prove the following result. Let H be a complex Hilbert space, let
$$
\mathcal{B}
$$
(H) be the algebra of all bounded linear operators on H, and let
$$
\mathcal{A}
$$
(H) ⊂
$$
\mathcal{B}
$$
(H) be a standard operator algebra which is closed under the adjoint operation. Suppose that T:
$$
\mathcal{A}
$$
(H) →
$$
\mathcal{B}
$$
(H) is a linear mapping satisfying T(AA* A) = T(A)A* A − AT(A*)A + AA*T(A) for all A ∈
$$
\mathcal{A}
$$
(H). Then T is of the form T(A) = AB + BA for all A ∈
$$
\mathcal{A}
$$
(H), where B is a fixed operator from
$$
\mathcal{B}
$$
(H). A result concerning functional equations related to bicircular projections is proved
Tài liệu tham khảo
K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, Inc. (New York, 1996).
M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc., 104 (1988), 1003–1006.
M. Brešar, Jordan mappings of semiprime rings, J. Algebra, 127 (1989), 218–228.
M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc., 37 (1988), 321–322.
J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc., 53 (1975), 321–324.
M. Fošner and D. Ilišević, On a class of projections on *-rings, Comm. Algebra, 33 (2005), 3293–3310.
M. Fošner and J. Vukman, On some equations in prime rings, preprint.
I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104–1119.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition American Mathematical Society (Providence, 1964).
L. Molnár, On centralizers of an H*-algebra, Publ. Math. Debrecen, 46 (1995), 89–95.
L. L. Stachó and B. Zalar, Bicircular projections on some matrix and operator spaces, Linear Algebra Appl., 384 (2004), 9–20.
L. L. Stachó and B. Zalar, Bicircular projections and characterization of Hilbert spaces, Proc. Amer. Math. Soc., 3 (2004), 3019–3025.
J. Vukman, On derivations of algebras with involution, Acta Math. Hungar., 112 (2006), 187–192.
J. Vukman, On functional equations related to bicircular projections, Glasnik Mat., 41 (2006), 51–56.
J. Vukman, I. Kosi-Ulbl and D. Eremita, On certain equations in rings, Bull. Austral. Math. Soc., 71 (2005), 53–60.