# ESSENTIAL NORMS AND STABILITY CONSTANTS OF WEIGHTED COMPOSITION OPERATORS ON C(X)

• Takagi, Hiroyuki ;
• Miura, Takeshi ;
• Takahasi, Sin-Ei
• Published : 2003.11.01
• 166 23

#### Abstract

For a weighted composition operator $uC_{\varphi}$ on C(X), we determine its essential norm and the constant for its Hyers-Ulam stability, in terms of the set $\varphi(\{x\;\in\;X\;:\;$\mid$u(x)$\mid$\;\geq\;r\})$ (r > 0).

#### Keywords

weighted composition operator;essential norm;Hyers-Ulam stability;closed range

#### References

1. Proc. Amer. Math. Soc. v.99 Compact and weakly compact composition operators on space of vector valued continuous functions R.K.Singh;W.H.Summers https://doi.org/10.1090/S0002-9939-1987-0877036-3
2. Bull. Korean Math. Soc. v.39 no.2 On the Hyers-Ulam stability of the Banach space-valued differential equation $\frak{y}^ '$ = $λ\frak{y}$ S.E.Takahasi;T.Miura;S.Miyajima https://doi.org/10.4134/BKMS.2002.39.2.309
3. J. Operator Theory v.19 Weighted composition operator on C(X, E) J.E.Jamison;M.Rajagopalan
4. Ann. of Math. v.125 no.2;2 The essential norm of a composition operator J.H.Shapiro https://doi.org/10.2307/1971314
5. Tokyo J. Math. v.24 On the Hyers-Ulam stability of real continuous function valued differentiable map T.Miura;S.E.Takahasi;H.Choda https://doi.org/10.3836/tjm/1255958187
6. Proc. Amer. Math. Soc. v.108 Compact weighted composition operators on Banach lattices W.Feldman https://doi.org/10.1090/S0002-9939-1990-0990422-6
7. Proc. Nat. Acad. Sci. U.S.A. v.27 On the stability of the linear functional equation D.H.Hyers https://doi.org/10.1073/pnas.27.4.222
8. Problems in modern mathematics S.M.Ulam
9. Introduction to functional analysis(2nd ed.) A.E.Taylor;D.C.Lay
10. Tokyo J. Math. v.11 Compact weighted composition operators on function algebras H.Takagi https://doi.org/10.3836/tjm/1270134266
11. Linear Operators Part Ⅰ N.Dunford;J.T.Schwartz
12. Math. Nachr. v.258 Hyers-Ulam stability of linear differential operator with constant coefficients T.Miura;S.Miyajima;S.E.Takahasi https://doi.org/10.1002/mana.200310088
13. Proc. Amer. Math. Soc. v.83 Compact weighted endomorphisms of C(X) H.Kamowitz
14. Composition operators on function spaces R.K.Singh;J.S.Manhas

#### Cited by

1. On the best constant in Hyers–Ulam stability of some positive linear operators vol.412, pp.1, 2014, https://doi.org/10.1016/j.jmaa.2013.10.039
2. Best constant in stability of some positive linear operators vol.90, pp.4, 2016, https://doi.org/10.1007/s00010-016-0405-3
3. Hyers–Ulam stability with respect to gauges vol.453, pp.1, 2017, https://doi.org/10.1016/j.jmaa.2017.04.022
4. Stability of some positive linear operators on compact disk vol.35, pp.6, 2015, https://doi.org/10.1016/S0252-9602(15)30070-9
5. Hyers–Ulam stability of linear functional differential equations vol.426, pp.2, 2015, https://doi.org/10.1016/j.jmaa.2015.02.018
6. Essential norm of substitution vector-valued integral operator on L1(Σ) space vol.448, pp.2, 2017, https://doi.org/10.1016/j.jmaa.2016.11.063
7. The Fréchet functional equation with application to the stability of certain operators vol.164, pp.1, 2012, https://doi.org/10.1016/j.jat.2011.09.009
8. Hyers–Ulam Stability of Differential Operators on Reproducing Kernel Function Spaces vol.10, pp.4, 2016, https://doi.org/10.1007/s11785-015-0486-3
9. On the stability of some positive linear operators from approximation theory vol.5, pp.2, 2015, https://doi.org/10.1007/s13373-015-0064-z
10. Hyers-Ulam Stability of Differentiation Operator on Hilbert Spaces of Entire Functions vol.2014, 2014, https://doi.org/10.1155/2014/398673
11. On the stability of some classical operators from approximation theory vol.31, pp.3, 2013, https://doi.org/10.1016/j.exmath.2013.01.007
12. Composition Operators on Cesàro Function Spaces vol.2014, 2014, https://doi.org/10.1155/2014/501057
13. Hyers-Ulam stability of delay differential equations of first order vol.289, pp.1, 2016, https://doi.org/10.1002/mana.201400298
14. Hyers–Ulam stability of impulsive integral equations pp.2198-2759, 2018, https://doi.org/10.1007/s40574-018-0180-2