# Some spectral results on ${L}^{2}\left({H}_{n}\right)$ related to the action of U(p,q)

Colloquium Mathematicae (2000)

- Volume: 86, Issue: 2, page 177-187
- ISSN: 0010-1354

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topGodoy, T., and Saal, L.. "Some spectral results on $L^{2}(H_{n})$ related to the action of U(p,q)." Colloquium Mathematicae 86.2 (2000): 177-187. <http://eudml.org/doc/210848>.

@article{Godoy2000,

abstract = {Let $H_\{n\}$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_\{n\}$. So $L^\{2\}(H_\{n\})$ has a natural structure of G-module. We obtain a decomposition of $L^\{2\}(H_\{n\})$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^\{2\}(H_\{n\})$ where $L=\sum _\{j=1\}^\{p\} (X_\{j\}^\{2\}+Y_\{j\}^\{2\}) - \sum _\{j=p+1\}^\{n\} (X_\{j\}^\{2\}+Y_\{j\}^\{2\})$, and where $\{X_\{1\},Y_\{1\},...,X_\{n\},Y_\{n\},T\}$ denotes the standard basis of the Lie algebra of $H_\{n\}$. Finally, we obtain a spectral characterization of the bounded operators on $L^\{2\}(H_\{n\})$ that commute with the action of G.},

author = {Godoy, T., Saal, L.},

journal = {Colloquium Mathematicae},

keywords = {spectral decomposition; joint spectrum; irreducible representation; Heisenberg group},

language = {eng},

number = {2},

pages = {177-187},

title = {Some spectral results on $L^\{2\}(H_\{n\})$ related to the action of U(p,q)},

url = {http://eudml.org/doc/210848},

volume = {86},

year = {2000},

}

TY - JOUR

AU - Godoy, T.

AU - Saal, L.

TI - Some spectral results on $L^{2}(H_{n})$ related to the action of U(p,q)

JO - Colloquium Mathematicae

PY - 2000

VL - 86

IS - 2

SP - 177

EP - 187

AB - Let $H_{n}$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_{n}$. So $L^{2}(H_{n})$ has a natural structure of G-module. We obtain a decomposition of $L^{2}(H_{n})$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^{2}(H_{n})$ where $L=\sum _{j=1}^{p} (X_{j}^{2}+Y_{j}^{2}) - \sum _{j=p+1}^{n} (X_{j}^{2}+Y_{j}^{2})$, and where ${X_{1},Y_{1},...,X_{n},Y_{n},T}$ denotes the standard basis of the Lie algebra of $H_{n}$. Finally, we obtain a spectral characterization of the bounded operators on $L^{2}(H_{n})$ that commute with the action of G.

LA - eng

KW - spectral decomposition; joint spectrum; irreducible representation; Heisenberg group

UR - http://eudml.org/doc/210848

ER -

## References

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