Annals of Functional Analysis

Geometry and operator theory on quaternionic Hilbert spaces

Bingzhe Hou and Geng Tian

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Abstract

In this article, we study the geometry and operator theory on quaternionic Hilbert spaces. As it is well-known, Cowen--Douglas operators are a class of non-normal operators related to complex geometry on complex Hilbert spaces. Our purpose is to generalize this concept on quaternionic Hilbert spaces. At the beginning, we study a class of complex holomorphic curves which naturally induce complex vector bundles as sub-bundles in the product space of the base space and a quaternionic Hilbert space. Then we introduce quaternionic Cowen--Douglas operators and give their quaternion unitarily equivalent invariant related to the geometry of the holomorphic curves.

Article information

Source
Ann. Funct. Anal., Volume 6, Number 4 (2015), 226-246.

Dates
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.afa/1435764014

Digital Object Identifier
doi:10.15352/afa/06-4-226

Mathematical Reviews number (MathSciNet)
MR3365994

Zentralblatt MATH identifier
1336.46041

Subjects
Primary: 46G20: Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]
Secondary: 47B99: None of the above, but in this section 51M15: Geometric constructions

Keywords
Cowen--Douglas operator quaternionic Hilbert space holomorphic curve unitary equivalence

Citation

Hou, Bingzhe; Tian, Geng. Geometry and operator theory on quaternionic Hilbert spaces. Ann. Funct. Anal. 6 (2015), no. 4, 226--246. doi:10.15352/afa/06-4-226. https://projecteuclid.org/euclid.afa/1435764014


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References

  • S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford U.P., New York, 1994.
  • F. Colombo, G. Gentili, I. Sabadini and D. Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math. 222 (2009), no. 5, 1793–1808.
  • F. Colombo, I. Sabadini and D. Struppa, Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions, Progress in Mathematics, Vol. 289, Springer Basel AG, Basel, 2011.
  • M.J. Cowen and R.G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 1, 187–261.
  • S. De Leo and G. Scolarici, Right eigenvalue equation in quaternionic quantum mechanics, J. Phys. A: Math. Gen. 33 (2000), no. 15, 2971–2995.
  • S. De Leo, G. Scolarici and L. Solombrino, Quaternionic eigenvalue problem, J. Math. Phys. 43 (2002), 5815–5829.
  • R. Feres and A. Zeghib, Leafwise holomorphic functions, Proc. Amer. Math. Soc. 131 (2003), 1717–1725.
  • D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Foundations of quaternion quantum mechanics, J. Math. Phys. 3 (1962), 207–220.
  • D. Finkelstein, J.M. Jauch and D. Speiser, Quaternionic representations of compact groups, J. Math. Phys. 4 (1963), 136–140.
  • C. Jiang and K. Ji, Similarity classification of holomorphic curves, Adv. Math. 215 (2007), 446–468.
  • S. Natarajan and K. Viswanath, Quaternionic representations of compact metric groups, J. Math. Phys. 8 (1967), 582–589.
  • C.S. Sharma and T.J. Coulson, Spectral theory for unitary operators on a quaternionic Hilbert space, J. Math. Phys. 28 (1987), no. 9, 1941–1946.
  • K. Viswanath, Normal Operators on Quaternionic Hilbert Spaces, Trans. Amer. Math. Soc. 162 (1971), 337–350.
  • F. Zhang, Quaternions and Matrices of Quaternions, Linear Algebra Appl. 251 (1997), 21–57.