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Dergiler / Turkish Journal of Mathematics / 2009 / Cilt: 33 - Sayı: 2

Perturbation of Closed Range Operators

Ghadir SADEGHİ
Sal Mohammad MOSLEHİAN
Sayfa
143–149
Yayın Tarihi
2022-11-04
DOI
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Thus we establish a link between T-bounded operators and Hyers--Ulam stability.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Let T, A be operators with domains D(T) \subseteq D(A) in a normed space X.
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Temel Bulgular & Veri:The operator A is called T-bounded if |Ax|\leq a|x|+b|Tx| for some a, b\geq 0 and all x \in D(T).
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Sonuç & Katkı:Thus we establish a link between T-bounded operators and Hyers--Ulam stability.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Hilbert space · perturbation · Hyers--Ulam stability

Özet

Let T, A be operators with domains D(T) \subseteq D(A) in a normed space X. The operator A is called T-bounded if |Ax|\leq a|x|+b|Tx| for some a, b\geq 0 and all x \in D(T). If A has the Hyers--Ulam stability then under some suitable assumptions we show that both T and S: = A+T have the Hyers--Ulam stability. We also discuss the best constant of Hyers--Ulam stability for the operator S. Thus we establish a link between T-bounded operators and Hyers--Ulam stability.

Abstract

Let T, A be operators with domains D(T) \subseteq D(A) in a normed space X. The operator A is called T-bounded if |Ax|\leq a|x|+b|Tx| for some a, b\geq 0 and all x \in D(T). If A has the Hyers--Ulam stability then under some suitable assumptions we show that both T and S: = A+T have the Hyers--Ulam stability. We also discuss the best constant of Hyers--Ulam stability for the operator S. Thus we establish a link between T-bounded operators and Hyers--Ulam stability.

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Bu Makaleye Atıf Yap

SADEGHİ, G., MOSLEHİAN, S. M. (2022). Perturbation of Closed Range Operators. Turkish Journal of Mathematics, Cilt: 33 - Sayı: 2, 143–149. https://ajindex.com/articles/608576

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Kaynakça (10)

  1. Conway, J.B.: A Course in Operator Theory, Graduate Studia in Mathemathics Volume 21, Amer. Math. Soc., Czerwik, S.: Functional equations and inequalities in several variables, World ScientiŞc, New Jersey, London, Singapore, Hong Kong, 2002. Goldberg S.: Unbounded linear Operators, McGraw-Hil, New Yourk, 1966.
  2. Hatori, O., Kobayashi, K., Miura, T., Takagi, H., Takahasi, S.E.: On the best constant of Hyers–Ulam stability, J. Nonlinear Convex Anal. 5, 387–393 (2004).
  3. Hirasawa, G., Miura, T. Hyers–Ulam Stability of a closed operator in a Hilbert space, Bull. Korean. Math. Soc. 43, –117 (2006).
  4. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables, Birkh¨auser, Basel, Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
  5. Kato, T.: Perturbation Theory for Linear Operators, Springer-Verlag,New York, 1966.
  6. Moslehian, M.S.: Ternary derivations, stability and physical aspects, Acta Applicandae Math. 100, no. 2, 187–199 (2008).
  7. Moslehian, M.S., Rassias, Th.M.: Stability of functional equations in non-Archimedian spaces, Appl. Anal. Disc. Math. 1, no. 2, 325–334 (2007).
  8. Rassias, Th.M.: On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62, no. 1, 23–130 (2000).
  9. Rassias, Th.M.: Functional Equations, Inequalities and Applications, Kluwer Academic Publishers,
  10. Dordrecht, Boston and London, 2003. Mohammad Sal MOSLEHIAN1,2, Ghadir SADEGHI1,2 Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box, Mashhad 91775, IRAN e-mail: [email protected] and [email protected] Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad-IRAN e-mail: [email protected]