Wandering Vectors and Frame Representations in Quaternionic Hilbert Spaces

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DOI: https://doi.org/10.28924/2291-8639-24-2026-367
Published: Oct 26, 2026
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Anu Kapoor, Manish Raghav, S.K. Sharma, Wandering Vectors and Frame Representations in Quaternionic Hilbert Spaces, Int. J. Anal. Appl., 24 (2026), 367.

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Anu Kapoor, Manish Raghav, S.K. Sharma

Abstract

We develop the theory of wandering vectors and frame representations in separable right quaternionic Hilbert spaces. We introduce the local commutant and obtain operator-theoretic characterizations of complete wandering vectors and normalized tight frame vectors. We prove that every complete normalized tight frame vector admits a dilation to a complete wandering vector in a larger quaternionic Hilbert space. Finally, we show that every quaternionic frame representation of a discrete group is unitarily equivalent to a subrepresentation of the left regular representation, yielding a canonical model for such representations.

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References

  1. D. Alpay, H.T. Kaptanoğlu, Quaternionic Hilbert Spaces and a von Neumann Inequality, Complex Var. Elliptic Equ. 57 (2012), 667–675. https://doi.org/10.1080/17476933.2010.534141.
  2. S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York, 1995. https://doi.org/10.1093/oso/9780195066432.001.0001.
  3. P.G. Casazza, The Art of Frame Theory, Taiwan. J. Math. 4 (2000), 129–201. https://doi.org/10.11650/twjm/1500407227.
  4. Q. Chen, P. Dang, T. Qian, A Frame Theory of Hardy Spaces with the Quaternionic and the Clifford Algebra Settings, Adv. Appl. Clifford Algebr. 27 (2017), 1073–1101. https://doi.org/10.1007/s00006-016-0736-0.
  5. O. Christensen, An Introduction to Frames and Riesz Bases, second ed., Birkhäuser, Cham, 2016. https://doi.org/10.1007/978-3-319-25613-9.
  6. F. Colombo, I. Sabadini, D.C. Struppa, Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions, Birkhäuser, Basel, 2011. https://doi.org/10.1007/978-3-0348-0110-2.
  7. R.J. Duffin, A.C. Schaeffer, A Class of Nonharmonic Fourier Series, Trans. Am. Math. Soc. 72 (1952), 341–366. https://doi.org/10.1090/S0002-9947-1952-0047179-6.
  8. R. Ghiloni, V. Moretti, A. Perotti, Continuous Slice Functional Calculus in Quaternionic Hilbert Spaces, Rev. Math. Phys. 25 (2013), 1350006. https://doi.org/10.1142/S0129055X13500062.
  9. X. Dai, D.R. Larson, Wandering Vectors for Unitary Systems and Orthogonal Wavelets, Mem. Am. Math. Soc. 134 (1998), no. 640. https://doi.org/10.1090/memo/0640.
  10. D. Han, D.R. Larson, Frames, Bases and Group Representations, Mem. Am. Math. Soc. 147 (2000), no. 697. https://doi.org/10.1090/memo/0697.
  11. C.-K. Ng, On Quaternionic Functional Analysis, Math. Proc. Camb. Philos. Soc. 143 (2007), 391–406. https://doi.org/10.1017/S0305004107000187.
  12. S.K. Sharma, S. Goel, Frames in Quaternionic Hilbert Spaces, Zh. Mat. Fiz. Anal. Geom. 15 (2019), 395–411. https://doi.org/10.15407/mag15.03.395.