# Cayley transform

The **Cayley transform** is a mapping that converts an unbounded self-adjoint operator on a [Hilbert space](https://www.edgepedia.org/hilbert-space) into a bounded one, in its operator form given by \( C_{A} = (A - iI)(A + iI)^{-1} \) for a linear operator \( A \) with dense domain, and in its matrix form a rational map that parametrizes orthogonal and unitary matrices by elements of the corresponding [Lie algebra](https://www.edgepedia.org/lie-algebra). A matrix version of the transform was considered by [Arthur Cayley](https://www.edgepedia.org/arthur-cayley), and the classical matrix form is described as a birational map between a quadratic matrix group and its Lie algebra first discovered by Cayley in 1846.<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup><sup> • </sup><sup>[2](https://arxiv.org/abs/2411.02071)</sup> In operator theory the transform is the central tool of von Neumann's theory of self-adjoint extensions, because it exchanges the properties of operators with spectra near the real line for properties of operators with spectra near the unit circle.<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup>

| Key fact | Detail |
|---|---|
| Operator definition | \( C_{A} = (A - iI)(A + iI)^{-1} \), defined on \( (A + iI)\,\mathrm{Dom}\,A \), for \( A \) with domain dense in a Hilbert space \( H \)<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup> |
| Inverse | \( A = i(I + T)(I - T)^{-1} \) recovers \( A \) from a transform \( T \) with \( \mathrm{Ker}(I - T) = \{0\} \)<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup> |
| Type correspondence | \( A \) is symmetric (self-adjoint) if and only if \( C_{A} \) is isometric (unitary)<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup> |
| Spectrum | \( \sigma(A) = \omega(\sigma(C_{A})) \) with \( \omega(\zeta) = i(1+\zeta)(1-\zeta)^{-1} \); \( A \) is bounded if and only if \( 1 \notin \sigma(C_{A}) \)<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup> |
| Extension criterion | A closed, densely defined symmetric operator has a self-adjoint extension if and only if the deficiency indices of its Cayley transform are equal<sup>[3](https://people.math.osu.edu/costin.9/7412-12/83.pdf)</sup> |
| Matrix form | A birational map between a quadratic matrix group and its Lie algebra, discovered by Cayley in 1846<sup>[2](https://arxiv.org/abs/2411.02071)</sup> |
| Matrix use | One of four standard parametrizations of orthogonal matrices by their minimal number of essential parameters<sup>[4](https://pubs.acs.org/doi/10.1021/acs.jpca.5b02015)</sup> |

## The operator transform

For a linear operator \( A \) whose domain is dense in a Hilbert space \( H \), the Cayley transform is the operator \( C_{A} = (A - iI)(A + iI)^{-1} \) with domain \( \mathrm{Dom}\,C_{A} = (A + iI)\,\mathrm{Dom}\,A \). The map establishes a correspondence between operators whose spectrum \( \sigma(A) \) lies close to the real line and operators whose spectrum is almost unitary, that is, close to the circle \( \{\zeta \in \mathbb{C} : |\zeta| = 1\} \).<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup>

The correspondence is exact for the standard operator classes. If \( A \) is a linear dissipative operator, then \( C_{A} \) is a contraction, meaning \( \|C_{A}x\| \le \|x\| \) for \( x \in \mathrm{Dom}\,A \), and \( \mathrm{Ker}(I - C_{A}) = \{0\} \). Conversely, if \( T \) is a contraction with \( \mathrm{Ker}(I - T) = \{0\} \) and \( (I - T)\,\mathrm{Dom}\,T \) dense in \( H \), then \( T = C_{A} \) for some linear dissipative operator \( A \), given by \( A = i(I + T)(I - T)^{-1} \).<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup> The key line of the correspondence runs through symmetry: A is symmetric exactly when C_A is isometric, and A is self-adjoint exactly when C_A is unitary.<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup>

When \( T \) is self-adjoint, the points \( \pm i \) lie in its resolvent set, so \( T + i \) and \( T - i \) are bijections of \( \mathrm{Dom}(T) \) onto \( H \) with bounded inverses, and \( C_{T} = (T - i)(T + i)^{-1} \) is a bounded operator defined on all of \( H \). In the resolvent convention \( R_{T}(z) = (z - T)^{-1} \), the transform can be written \( C_{T} = I + 2iR_{T}(-i) \).<sup>[5](https://alphabetamath.cc/functional-analysis/unbounded-self-adjoint-operators-and-stones-theorem)</sup>

The spectrum behaves predictably under the map: \( \sigma(A) = \omega(\sigma(C_{A})) \), where \( \omega(\zeta) = i(1+\zeta)(1-\zeta)^{-1} \). In particular, \( A \) is bounded if and only if \( 1 \notin \sigma(C_{A}) \). The transform is compatible with operator ideals as well: given an operator ideal \( \gamma \) in \( H \), membership of \( A - B \) in \( \gamma \) entails membership of \( C_{A} - C_{B} \) in \( \gamma \), and when \( A, B \) are bounded this implication can be reversed.<sup>[1](https://encyclopediaofmath.org/index.php?title=Cayley_transform)</sup>

## Von Neumann's extension theory

The transform's most consequential application is [John von Neumann](https://www.edgepedia.org/john-von-neumann)'s characterization of self-adjoint extensions. For a closed, densely defined symmetric operator \( T \), the Cayley transform \( U = (T - i)(T + i)^{-1} \) is an isometry mapping one closed subspace, written \( H^{+} \), onto another, \( H^{-} \); the dimensions of these two deficiency subspaces are the deficiency indices of \( T \).<sup>[3](https://people.math.osu.edu/costin.9/7412-12/83.pdf)</sup><sup> • </sup><sup>[6](https://nbow.io/de/products/ai-powered-research/treatise-of-physics/chapter/appendices/26-vonneumann-extensions.html)</sup>

**Von Neumann's theorem.** A closed, densely defined symmetric operator \( T \) has a self-adjoint extension if and only if the deficiency indices of its Cayley transform are equal.<sup>[3](https://people.math.osu.edu/costin.9/7412-12/83.pdf)</sup> The proof works by converting operators into isometries: if \( \tilde{T} \) is a self-adjoint extension of \( T \), its Cayley transform \( \tilde{U} = (\tilde{T} - i)(\tilde{T} + i)^{-1} \) is unitary and extends \( U \), which forces the two deficiency spaces to have the same dimension. Conversely, when \( H^{+} \) and \( H^{-} \) have the same dimension, \( U \) admits a unitary extension \( \tilde{U} \), and \( \tilde{T} = i(1 + \tilde{U})(1 - \tilde{U})^{-1} \) is a self-adjoint operator extending \( T \).<sup>[3](https://people.math.osu.edu/costin.9/7412-12/83.pdf)</sup>

The full classification is sharper. For a closed symmetric operator, self-adjointness holds precisely in the case where both deficiency indices are zero, and self-adjoint extensions exist precisely when the two indices coincide; when that happens, the extensions correspond one-to-one with the unitary mappings from one deficiency subspace to the other.<sup>[6](https://nbow.io/de/products/ai-powered-research/treatise-of-physics/chapter/appendices/26-vonneumann-extensions.html)</sup> The reduction of the extension problem to counting dimensions is the reason the method works: the Cayley transform turns the extension problem for operators into the extension problem for isometries, where it becomes a matter of comparing dimensions.<sup>[6](https://nbow.io/de/products/ai-powered-research/treatise-of-physics/chapter/appendices/26-vonneumann-extensions.html)</sup> The criterion also constrains examples: a symmetric operator whose deficiency indices are both one has, in effect, a one-parameter family of self-adjoint extensions, a situation that arises in the analysis of differential operators such as the Laplace-Beltrami operator.<sup>[7](https://www.math.ubc.ca/~feldman/m511/cayley.pdf)</sup><sup> • </sup><sup>[8](https://www.researchgate.net/publication/255722499_On_the_Theory_of_Self-Adjoint_Extensions_of_the_Laplace-Beltrami_Operator_Quadratic_Forms_and_Symmetry)</sup>

The transform also respects symmetry reduction: a subspace reduces a symmetric operator \( X \) if and only if it reduces its Cayley transform, which lets extension questions be studied on reducing subspaces.<sup>[9](https://citeseerx.ist.psu.edu/document?doi=91eeb10db8dcea86d12bfbccaae5a899d916cb9b&repid=rep1&type=pdf)</sup> In Kreĭn's parallel theory of self-adjoint extensions of positive operators, the Kreĭn transform plays a role that parallels the role of the Cayley transform in von Neumann's theory.<sup>[10](https://www.researchgate.net/publication/369815033_Classical_Self-Adjoint_Extension_Schemes)</sup>

## The matrix transform and group parametrization

In its original matrix setting, the Cayley transform is a birational map between a quadratic matrix group and its Lie algebra, first discovered by Cayley in 1846.<sup>[2](https://arxiv.org/abs/2411.02071)</sup> Arthur Cayley (1821–1895) is also credited with founding the theory of matrices through the first expository articles on the subject, published after [James Joseph Sylvester](https://www.edgepedia.org/james-joseph-sylvester) introduced the term "matrix" in 1850.<sup>[11](https://mathshistory.st-andrews.ac.uk/Biographies/Cayley/)</sup>

In a comparative study of representations of orthogonal matrices, the rational Cayley transform representation appears alongside the exponential, Householder reflector, and [Givens rotation](https://www.edgepedia.org/givens-rotation) representations as one of four parametrizations by the minimal number of essential parameters, with the practical issues of numerical stability, continuity, and uniqueness discussed for each.<sup>[4](https://pubs.acs.org/doi/10.1021/acs.jpca.5b02015)</sup>

The map is local rather than global: it fails where \( I - T \) is not invertible, which corresponds to the eigenvalue 1. For the special unitary group \( \mathrm{SU(3)} \), a modified Cayley transform has been proposed to define a suitable local parameterization, with particular attention to the hybrid [Monte Carlo algorithm](https://www.edgepedia.org/monte-carlo-algorithm) for gauge field generation in lattice quantum chromodynamics.<sup>[12](https://arxiv.org/abs/2406.11337)</sup>

## Wider use

Because the Cayley transform converts unbounded self-adjoint operators into bounded unitary ones, spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through it, and bounded-transform approaches extend this functional calculus to operators affiliated with von Neumann algebras.<sup>[13](https://arxiv.org/abs/1508.06772)</sup> The same mechanism appears in applied work: self-adjoint extension schemes for singular Hamiltonians in quantum mechanics are organized around the von Neumann–Kreĭn theory in which the transform supplies the extension classification.<sup>[14](https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2019.00175/full)</sup>

## References

1. [Cayley transform, Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=Cayley_transform)
2. [The Cayley Transform on Representations, arXiv:2411.02071](https://arxiv.org/abs/2411.02071)
3. [Von Neumann's Theorem on self-adjoint extensions, lecture notes, Ohio State University (Math 7412)](https://people.math.osu.edu/costin.9/7412-12/83.pdf)
4. [The Representation and Parametrization of Orthogonal Matrices, Journal of Physical Chemistry A](https://pubs.acs.org/doi/10.1021/acs.jpca.5b02015)
5. [Unbounded Self Adjoint Operators and Stone's Theorem, Alphabeta Math](https://alphabetamath.cc/functional-analysis/unbounded-self-adjoint-operators-and-stones-theorem)
6. [Von Neumann's Criterion for Self-Adjoint Extensions, Treatise of Physics](https://nbow.io/de/products/ai-powered-research/treatise-of-physics/chapter/appendices/26-vonneumann-extensions.html)
7. [The Cayley Transform and Self-adjoint Extensions, UBC Math 511 notes (S. Feldman)](https://www.math.ubc.ca/~feldman/m511/cayley.pdf)
8. [On the Theory of Self-Adjoint Extensions of the Laplace-Beltrami Operator, Quadratic Forms and Symmetry](https://www.researchgate.net/publication/255722499_On_the_Theory_of_Self-Adjoint_Extensions_of_the_Laplace-Beltrami_Operator_Quadratic_Forms_and_Symmetry)
9. [ufift-hep-99-08](https://citeseerx.ist.psu.edu/document?doi=91eeb10db8dcea86d12bfbccaae5a899d916cb9b&repid=rep1&type=pdf)
10. [Classical Self-Adjoint Extension Schemes, in Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians](https://www.researchgate.net/publication/369815033_Classical_Self-Adjoint_Extension_Schemes)
11. [Arthur Cayley (1821–1895), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Cayley/)
12. [A modified Cayley transform for SU(3), arXiv:2406.11337](https://arxiv.org/abs/2406.11337)
13. [A bounded transform approach to self-adjoint operators, arXiv:1508.06772](https://arxiv.org/abs/1508.06772)
14. [Self-Adjoint Extension Approach for Singular Hamiltonians in (2+1) Dimensions, Frontiers in Physics](https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2019.00175/full)

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