### Abstract

In this paper we show that the separable representation theory of U(H) is completely analogous to that for U(C^{)n}, in that every separable representation is discretely decomposable and the irreducible representations all occur in the decomposition of the mixed tensor algebra of H. This was previously shown to be true (for all representations, separable and nonseparable) for the normal subgroup U(if)∞, consisting of operators which are compact perturbations of the identity, by Kirillov and Ol'shanskii. In particular we show that all nontrivial representations of the unitary Calkin group are nonseparable. The proof exploits the analogue of the following fact about the Calkin algebra: if π is a nontrivial representation of the Calkin algebra and T is a normal operator on H, then every point in the spectrum of π(T) is an eigenvalue.

Original language | English (US) |
---|---|

Pages (from-to) | 416-420 |

Number of pages | 5 |

Journal | Proceedings of the American Mathematical Society |

Volume | 102 |

Issue number | 2 |

DOIs | |

State | Published - 1988 |

### Fingerprint

### Keywords

- Calkin algebra
- Separable representation
- Spectrum
- Unitary group

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**The Separable Representations Of U(H).** / Pickrell, Douglas M.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 102, no. 2, pp. 416-420. https://doi.org/10.1090/S0002-9939-1988-0921009-X

}

TY - JOUR

T1 - The Separable Representations Of U(H)

AU - Pickrell, Douglas M

PY - 1988

Y1 - 1988

N2 - In this paper we show that the separable representation theory of U(H) is completely analogous to that for U(C)n, in that every separable representation is discretely decomposable and the irreducible representations all occur in the decomposition of the mixed tensor algebra of H. This was previously shown to be true (for all representations, separable and nonseparable) for the normal subgroup U(if)∞, consisting of operators which are compact perturbations of the identity, by Kirillov and Ol'shanskii. In particular we show that all nontrivial representations of the unitary Calkin group are nonseparable. The proof exploits the analogue of the following fact about the Calkin algebra: if π is a nontrivial representation of the Calkin algebra and T is a normal operator on H, then every point in the spectrum of π(T) is an eigenvalue.

AB - In this paper we show that the separable representation theory of U(H) is completely analogous to that for U(C)n, in that every separable representation is discretely decomposable and the irreducible representations all occur in the decomposition of the mixed tensor algebra of H. This was previously shown to be true (for all representations, separable and nonseparable) for the normal subgroup U(if)∞, consisting of operators which are compact perturbations of the identity, by Kirillov and Ol'shanskii. In particular we show that all nontrivial representations of the unitary Calkin group are nonseparable. The proof exploits the analogue of the following fact about the Calkin algebra: if π is a nontrivial representation of the Calkin algebra and T is a normal operator on H, then every point in the spectrum of π(T) is an eigenvalue.

KW - Calkin algebra

KW - Separable representation

KW - Spectrum

KW - Unitary group

UR - http://www.scopus.com/inward/record.url?scp=84966244104&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966244104&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1988-0921009-X

DO - 10.1090/S0002-9939-1988-0921009-X

M3 - Article

VL - 102

SP - 416

EP - 420

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -