Abstract
The notion of Cauchy dual for left-invertible covariant representations was studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we extend this notion for the covariant representations having closed range and explore several useful properties. We obtain a Wold-type decomposition for regular completely bounded covariant representation whose Moore-Penrose inverse is regular. Also, we discuss an example related to the non-commutative bilateral weighted shift. We prove that the Cauchy dual of the concave covariant representation \((\sigma , V)\) modulo \(N(\widetilde{V})\) is hyponormal modulo \(N(\widetilde{V})\).
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References
Aleman, A.: The multiplication operators on Hilbert spaces of analytic functions. Habilitationsschrift, Fernuversität Hagen (1993)
Athavale, A.: On completely hyperexpansive operators. Proc. Am. Math. Soc. 124, 3745–3752 (1996)
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses Theory and Applications. Canadian Mathematic Society, 2nd edn. Springer, New York (2003)
Beurling, A.: On two problems concerning linear transformations in Hilbert space. Acta Math. 81, 239–255 (1949)
Chavan, S.: On operators Cauchy dual to 2-hyperexpansive operators. Proc. Edinb. Math. Soc. 50, 637–652 (2007)
Cuntz, J.: Simple \(C^*\)-algebras generated by isometries. Commun. Math. Phys. 57(2), 173–185 (1977)
Ezzahraoui, H., Mbekhta, M., Zerouali, E.H.: Wold-type decomposition for some regular operators. J. Math. Anal. Appl. 430(1), 483–499 (2015)
Ezzahraoui, H., Mbekhta, M., Zerouali, E.H.: On the Cauchy dual of closed range operators. Acta Sci. Math. (Szeged) 85, 231–248 (2019)
Ezzahraoui, H., Mbekhta, M., Zerouali, E.H.: Wold-type decomposition for bi-regular operators. Acta Sci. Math. (Szeged) 87, 463–483 (2021)
Fowler, N.J.: Discrete product systems of Hilbert bimodules. Pacific J. Math. 204(2), 335–375 (2002)
Frazho, A.E.: Complements to models for noncommuting operators. J. Funct. Anal. 59, 445–461 (1984)
Groetsch, C.W.: Generalized inverses of linear operators: representation and approximation, Monographs and Textbooks in Pure and Applied Mathematics, No. 37. Marcel Dekker, Inc. New York (1977)
Guyker, J.: A structure theorem for operators with closed range. Bull. Austral. Math. Soc. 18, 169–186 (1978)
Halmos, P.: Shifts on Hilbert spaces. J. Reine Angew. Math. 208, 102–112 (1961)
Lance, E.C.: Hilbert \(C^*\)-Modules, London Mathematical Society Lecture Note Series, vol. 210. Cambridge University Press, Cambridge (1995)
Muhly, P.S., Solel, B.: Tensor algebras over \(C^*\)-correspondences: representations, dilations, and \(C^*\)-envelopes. J. Funct. Anal. 158(2), 389–457 (1998)
Muhly, P.S., Solel, B.: Tensor Algebras, Induced Representations, and the Wold Decomposition. Can. J. Math. 51(4), 850–880 (1999)
Olofsson, A.: Wandering subspace theorems. Integral Equ. Oper. Theory 51, 395–409 (2005)
Pimsner, M.V.: A class of \(C^*\)-algebras generalizing both Cuntz-Krieger algebras and crossed products by \({\bf Z}\), Free probability theory (Waterloo, ON, 1995), pp. 189–212, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI (1997)
Popescu, G.: Isometric dilations for infinite sequences of noncommuting operators. Trans. Am. Math. Soc. 319(2), 523–536 (1989)
Richter, S.: Invariant subspaces of the Dirichlet shift. J. Reine Angew. Math. 386, 205–220 (1988)
Rohilla, A., Veerabathiran, S., Trivedi, H.: Regular covariant representations and their Wold-type decomposition. J. Math. Anal. Appl. 522(2), 126997 (2023)
Shimorin, S.: Wold-type decompositions and wandering subspaces for operators close to isometries. J. Reine Angew. Math. 531, 147–189 (2001)
Sutton, D.J.: Structure of invariant subspaces for left-invertible operators on Hilbert Space. Ph.D. thesis, Faculty of the Virginia Polytechnic Institute and State University, Blacksburg, Virginia (2010)
Trivedi, H., Veerabathiran, S.: Generating wandering subspaces for doubly commuting covariant representations. Integr. Equ. Oper. Theory 91(4), 35 (2019)
Wold, H.: A Study in the Analysis of Stationary Time Series. Almquist and Wiksell, Uppsala (1938)
Acknowledgements
The author is grateful to the reviewer for carefully reading the manuscript and giving valuable suggestions and comments. He want to thank Harsh Trivedi and Shankar Veerabathiran for some fruitful discussions. The author supported by UGC fellowship (File No:16-6(DEC. 2018)/2019(NET/CSIR)) and acknowledge the Centre for Mathematical & Financial Computing and the DST-FIST grant for the financial support for the computing lab facility under the scheme FIST (File No: SR/FST/MS-I/2018/24) at the LNMIIT, Jaipur.
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Saini, D. Cauchy dual and Wold-type decomposition for bi-regular covariant representations. Acta Sci. Math. (Szeged) 90, 123–144 (2024). https://doi.org/10.1007/s44146-023-00105-7
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DOI: https://doi.org/10.1007/s44146-023-00105-7