Abstract
We introduce the C-decomposition property for reducible bounded linear operators on a Hilbert space, and prove that an arbitrary idempotent operator has the C-decomposition property with respect to a particular space decomposition, which is related to Halmos’ two projections theory. Using this, we obtain a general explicit description for all the conjugations C such that a given idempotent operator is a C-projection. We also present a characterization of the ranges of C-projections for any conjugation C.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
All data included in this study are available.
References
Benhida, C., Chō, M., Ko, E., Lee, J.E.: On the generalized mean transforms of complex symmetric operators. Banach J. Math. Anal. 14, 842–855 (2020)
Böttcher, A., Spitkovsky, I.M.: A gentle guide to the basics of two projections theory. Linear Algebra Appl. 432, 1412–1459 (2010)
Deng, C.Y., Du, H.K.: Common complements of two subspaces and an answer to Groß’s question. Acta Math. Sin. (Chin. Ser.) 49, 1099–1112 (2006)
Garcia, S.R.: Aluthge transforms of complex symmetric operators. Integr. Equ. Oper. Theory 60, 357–367 (2008)
Garcia, S.R., Prodan, E., Putinar, M.: Mathematical and physical aspects of complex symmetric operators. J. Phys. A: Math. Theor. 47, 353001 (2014)
Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358, 1285–1315 (2006)
Garcia, S.R., Putinar, M.: Complex symmetric operators and applications II. Trans. Am. Math. Soc. 359, 3913–3931 (2007)
Garcia, S.R., Wogen, W.R.: Complex symmetric partial isometries. J. Funct. Anal. 257, 1251–1260 (2009)
Garcia, S.R., Wogen, W.R.: Some new classes of complex symmetric operators. Trans. Am. Math. Soc. 362, 6065–6077 (2010)
Guo, K.Y., Ji, Y.Q., Zhu, S.: A C\(^{*}\)-algebra approach to complex symmetric operators. Trans. Am. Math. Soc. 367, 6903–6942 (2015)
Guo, K.Y., Zhu, S.: A canonical decomposition of complex symmetric operators. J. Oper. Theory 72, 529–547 (2014)
Halmos, P.: Two subspaces. Trans. Am. Math. Soc. 144, 381–389 (1969)
Liu, T., Shi, L.Y., Wang, C., Zhu, S.: An interpolation problem for conjugations. J. Math. Anal. Appl. 500, 125118 (2021)
Liu, T., Zhao, J.Y., Zhu, S.: Reducible and irreducible approximation of complex symmetric operators. J. Lond. Math. Soc. 100, 341–360 (2019)
Xu, X.M., Li, Y.: Symmetries for \(J\)-projection. Anal. Math. 46, 393–407 (2020)
Zhu, S., Li, C.G., Ji, Y.Q.: The class of complex symmetric operators is not norm closed. Proc. Am. Math. Soc. 140, 1705–1708 (2012)
Acknowledgements
This work was supported by NSF of China (Nos. 11601339, 11671242, 11771401, 12271134).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hugo Woerdeman.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xu, XM., Yuan, Y., Li, Y. et al. On conjugations concerning idempotents. Ann. Funct. Anal. 15, 52 (2024). https://doi.org/10.1007/s43034-024-00354-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43034-024-00354-9