## Linear Operators: Spectral theory |

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Page 868

If 3 is a closed ideal in the commutative B-algebra X then the quotient algebra X/

S is isometrically isomorphic to

maximal. Proof. If 9 is not maximal it is properly contained in an ideal and so X/Q1

...

If 3 is a closed ideal in the commutative B-algebra X then the quotient algebra X/

S is isometrically isomorphic to

**the field**of complex numbers if and only if Of ismaximal. Proof. If 9 is not maximal it is properly contained in an ideal and so X/Q1

...

Page 932

114], and similar results for semigroups of contractions are found in Sz.-Nagy [8,

10] (see also Cooper [3]). Sz.-Nagy's original proof depended on the following

theorem due to Neumark [3]. Theorem. Let S be an abstract set and E

resp.

114], and similar results for semigroups of contractions are found in Sz.-Nagy [8,

10] (see also Cooper [3]). Sz.-Nagy's original proof depended on the following

theorem due to Neumark [3]. Theorem. Let S be an abstract set and E

**a field**(resp.

Page 1153

Since

integration as developed in Chapter III may be used as

Thus

Since

**the**measure space (R, 27, X) is**a**or-finite measure space**the**theory**of**integration as developed in Chapter III may be used as

**a**basis for**the**theory ...Thus

**the**product group has**a**Haar measure**A**(2) defined on its Borel**field**27<2).### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero