## Linear Operators: Spectral theory |

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Page 1145

The Peter- Weyl Theorem 1.4 is basic to the theory of

groups. The principal definitions and theorems of this theory are as follows.

Definition: Let G be a topological group, and X a fi-space. Then a

R of ...

The Peter- Weyl Theorem 1.4 is basic to the theory of

**representations**of compactgroups. The principal definitions and theorems of this theory are as follows.

Definition: Let G be a topological group, and X a fi-space. Then a

**representation**R of ...

Page 1146

Any finite dimensional

irreducible

dimensional

generality, confine ...

Any finite dimensional

**representation**of a compact group G is a direct sum ofirreducible

**representations**. This theorem shows that in studying finitedimensional

**representations**of a compact group G we may, without loss ofgenerality, confine ...

Page 1217

fin(e) = fx(e n ej, e e n = 1, 2, A spectral

LiL2(/in) relative to a self adjoint operator T in ,£> is said to be an ordered

ordered ...

fin(e) = fx(e n ej, e e n = 1, 2, A spectral

**representation**of a Hilbert space onto ^LiL2(/in) relative to a self adjoint operator T in ,£> is said to be an ordered

**representation**of § relative to T. The measure /< is called the measure of theordered ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic B-algebra 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero