Approximate n -idempotents and generalized Aluthge transform
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Aequationes Mathematicae
Approximate n-idempotents and generalized Aluthge transform Mohammad Sal Moslehian Dedicated to Professor Asadollah Niknam on his 70th birthday with respect and affection. Abstract. Let p be a real number and let ε > 0. An operator T ∈ B(H ) is called a (p, ε)approximate n-idempotent if T n x − T x ≤ εxp
(x ∈ H ) .
In this note, we remark that if p = 1, then T is an n-idempotent. If p = 1, the operator T is √ , then there is a self-adjoint a self-adjoint contraction satisfying (−T )n ≥ 0, and ε < n n−1 n−1 n
n-idempotent S such that T − S < Kε for some constant K > 0. Among other results, we examine the lack of a similar result for the (1, ε)-approximate generalized Aluthge transform. Mathematics Subject Classification. 47A55, 39B82, 47B15. Keywords. Generalized Aluthge transform, n-idempotent, Quasinormal operator, Stability.
1. Introduction An interesting problem in the theory of functional and operator equations is “When is it true that a mapping which satisfies a functional or operator equation approximately must be close to an exact solution of the functional or operator equation?” Such a problem was first proposed by Ulam [15] as follows: “Given a group G1 , a metric group (G2 , d) and a positive number ε, does there exist a δ > 0 such that if a mapping f : G1 → G2 satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G1 , then there exists a homomorphism T : G1 → G2 such that d(f (x), T (x)) < ε for all x ∈ G1 ?” Hyers [6] partially solved the problem in the framework of Banach spaces with δ = ε as follows:
M. S. Moslehian
AEM
“Suppose that E1 , E2 are Banach spaces and f : E1 → E2 is a mapping for which there exists ε > 0 such that f (x + y) − f (x) − f (y) < ε for all x, y ∈ E1 . Then there is a unique additive mapping T : E1 → E2 such that f (x) − T (x) < ε for all x ∈ E1 .” This topic is closely related to the perturbation of operators, which has been extensively investigated by many mathematicians; see [3,8,12,16]. In perturbation theory, several situations occur: (1) There is a problem that we do not know the exact solution of, but we can solve a slightly different problem; (2) we have a mathematical object approximately satisfying a property, and we try to show that it is close to a mathematical object exactly satisfying that property. For instance, if A and B are two subalgebras of a Banach algebra C such that A and B are “geometrically close”, then it is interesting to investigate whether A and B are isomorphic; see the interesting Lecture Note [8] of Jarosz. Miura et al. [12] explored approximate normal operators. Mirzavaziri et al. [11] studied (p, )-approximate operators. Maher and Moslehian [10] investigated approximate partial isometries and approximate generalized inverses. In this paper, we deal with (p, ε)-approximate n-idempotents and provide several properties of such operators. We show that if p = 1, the operator T is √ a self-adjoint contraction satisfying (−T )n ≥ 0, and ε < n n−1 n−1 n , then there is a self-adjoint n-idempotent S su
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