Abstract
In this article, we briefly describe nearly \(T^{-1}\) invariant subspaces with finite defect for a shift operator T having finite multiplicity acting on a separable Hilbert space \({\mathcal {H}}\) as a generalization of nearly \(T^{-1}\) invariant subspaces introduced by Liang and Partington in Complex Anal. Oper. Theory 15(1) (2021) 17 pp. In other words, we characterize nearly \(T^{-1}\) invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in Int. Equations Oper. Theory 92 (2020) 1–15. Furthermore, we also provide a concrete representation of the nearly \(T_B^{-1}\) invariant subspaces with finite defect in a scale of Dirichlet-type spaces \({\mathcal {D}}_\alpha \) for \(\alpha \in [-1,1]\) corresponding to any finite Blashcke product B, as was done recently by Liang and Partington for defect zero case (see Section 3 of Complex Anal. Oper. Theory 15(1) (2021) 17 pp).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brown L and Shields A L, Cyclic Vectors in the Dirichlet Space, Trans. Amer. Math. Soc. 285 (1984)
Chalendar I, Chevrot N and Partington J R, Nearly invariant subspaces for backwards shifts on vector-valued Hardy spaces, J. Oper. Theory 63(2) (2010) 403–41
Chalendar I, Gallardo-Gutiérrez E A and Partington J R, Weighted composition operators on the Dirichlet space: boundedness and spectral properties, Math. Ann. 363 (2015) 1265–1279
Chalendar I, Gallardo-Gutiérrez E A and Partington J R, A Beurling theorem for almost-invariant subspaces of the shift operator, J. Oper. Theory 83 (2020) 321–331
Chattopadhyay A, Das S and Pradhan C, Almost invariant subspaces of the shift operator on vector-valued Hardy spaces, Integral Equations Oper. Theory 92 (2020) 1–15
Conway J B, A Course in Operator Theory, American Mathematical Society, Providence (2000)
El-Fallah O, Kellay K, Mashreghi J and Ransford T, A primer on the Dirichlet space, Cambridge Tracts in Mathematics, vol. 203 (2014) (Cambridge: Cambridge University Press)
Erard C, Nearly invariant subspaces related to multiplication operators in hilbert spaces of analytic functions, Integral Equations Oper. Theory 50 (2004) 197–210
Fricain E and Mashreghi J, The theory of \(H(b)\) spaces, Vol. 1, New Mathematical Monographs, vol. 20 (2016) (Cambridge: Cambridge University Press)
Fricain E and Mashreghi J, The theory of \(H(b)\) spaces, Vol. 2, New Mathematical Monographs, vol. 21 (2016) (Cambridge: Cambridge University Press)
Gallardo-Gutiérrez E A, Partington J R and Seco D, On the Wandering Property in Dirichlet Spaces, Integral Equations Oper. Theory 92 (2020) 16
Garcia S R, Mashreghi J and Ross W, Finite Blaschke Products and their Connections (2018) (New York: Springer)
Hayashi E, The kernel of a Toeplitz operator, Integral Equations Oper. Theory 9(4) (1986) 588–591
Hitt D, Invariant subspaces of \(H^2\) of an annulus, Pacific J. Math. 134(1) (1988) 101–120
Lance T L and Stessin M I, Multiplication invariant subspaces of Hardy spaces, Can. J. Math. 49(1) (1997) 100–118
Liang Y and Partington J R, Nearly invariant subspaces for operators in Hilbert spaces, Complex Anal. Oper. Theory 15(1) (2021) Paper No. 5, 17 pp.
Martínez-Avendaño R A and Rosenthal P, An introduction to operators on the Hardy–Hilbert space, Graduate Texts in Mathematics, vol. 237 (2007) (New York: Springer)
Nikolski N, Operators, functions, and systems: an easy reading Vol. 1 Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI (2002)
O’Loughlin R, Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators, Complex Anal. Oper. Theory 14(8) (2020) 10-24
Partington J R, Linear operators and linear systems. An analytical approach to control theory, London Mathematical Society Student Texts, vol. 60 (2004) (Cambridge: Cambridge University Press)
Popov A and Tcaciuc A, Every operator has almost-invariant subspaces, J. Funct. Anal. 265(2) (2013) 257–265
Rosenblum M and Rovnyak J, Hardy classes and operator theory (1985) (New York: Oxford University Press)
Sarason D, Nearly invariant subspaces of the backward shift, Contributions to Operator Theory and its Applications (Mesa, AZ, 1987), pp. 481–493, Oper. Theory Adv. Appl., vol. 35 (1988) (Basel: Birkhäuser)
Sz-Nagy B and Foiaş C, Harmonic Analysis of Operators on Hilbert Space (1970) (Amsterdam: North Holland)
Taylor G R, Multipliers on \({\cal{D}}_\alpha \), Trans. Amer. Math. Soc. 123 (1966) 229–240
Acknowledgements
The research of the first-named author is supported by the Mathematical Research Impact Centric Support (MATRICS) grant, File No: MTR/2019/000640, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. The second-named author gratefully acknowledge the support provided by IIT Guwahati, Government of India.
Author information
Authors and Affiliations
Additional information
Communicated by B V Rajarama Bhat.
Rights and permissions
About this article
Cite this article
Chattopadhyay, A., Das, S. Study of nearly invariant subspaces with finite defect in Hilbert spaces. Proc Math Sci 132, 10 (2022). https://doi.org/10.1007/s12044-022-00654-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12044-022-00654-x
Keywords
- Vector-valued Hardy space
- nearly invariant subspaces with finite defect
- multiplication operator
- Beurling’s theorem
- Dirichlet space
- Blaschke products