Document Type : Research Paper
Authors
- Noha Eftekhari ^{} ^{}
- Ali Bayati Eshkaftaki
Department of pure Mathematics, Faculty of Mathematical Sciences, University of Shahrekord, P.O.Box 115, Shahrekord, 88186-34141, Iran.
Abstract
The aim of this work is to characterize all bounded linear operators $T:\lpi\rightarrow\lpi$ which preserve disjoint support property. We show that the constant coefficients of all isometries on $\lpi$ are in the class of such operators, where $2\neq p\in [1,\infty )$ and $I$ is a non-empty set. We extend preserving disjoint support property to linear operators on $\mathfrak{c}_{0}(I).$ At the end, we obtain some equivalent properties of isometries on Banach spaces.
Keywords
[2] Y.A. Abramovich, A.I. Veksler and A.V. Koldunov, Operators preserving disjointness, Dokl. Akad. Nauk USSR, 248 (1979), pp. 1033-1036.
[3] T. Ando, Majorization and inequalities in matrix theory, Linear Algebra Appl., 199 (1994), pp. 17-67.
[4] F. Bahrami, A. Bayati and S.M. Manjegani, Linear preservers of majorization on $lpi$, Linear Algebra Appl., 436 (2012), pp. 3177-3195.
[5] S. Banach, Theorie des operations lineares, Chelsea, Warsaw, 1932.
[6] N.L. Carothers, A Short course on Banach space theory, Cambridge University Press, 2005.
[7] J.T. Chan, Operators with the disjoint support property, J. Operator Theory, 24 (1990), pp. 383-391.
[8] R.J. Fleming and J.E. Jamison, Isometries on Banach spaces: vector-valued function spaces, Vol. 2, Taylor and Francis Group, LLC, 2008.
[9] H-L. Gau, J-S. Jeang and N-C. Wong, Biseparating linear maps between continuous vector-valued function spaces, J. Aust. Math. Soc., 74 (2003), pp. 101-109.
[10] J-S. Jeang and N-C. Wong, Weighted composition operators of $C_0(X)$'s, J. Math. Anal. Appl., 201 (1996), pp. 981-993.
[11] A. Jimenez-Vergas and Ya-Shu Wang, Linear biseparating maps between vector-valued little Lipschitz function spaces, Acta Math. Sin. (Engl. Ser. ), 26 (2010), pp. 1005-1018.
[12] J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math., 8 (1958), pp. 459-466.
[13] S. Pierce, A survey of linear preserver problems, Linear Multilinear Algebra, 33 (1992), pp. 1-2.
[14] B.Z. Volkh, On linear multiplicative operations, Dokl. Akad. Nauk USSR, 41 (1943), pp. 148-151.
[15] H. Zhang, Orthogonality from disjoint support in reproducing kernel Hilbert spaces, J. Math. Anal. Appl., 349 (2009), pp. 201-210.