## Linear Operators: Spectral theory |

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Page 950

Every such group has a non-negative countably additive

defined on the Borel sets E, finite on compact sets, positive or infinite on open

sets, invariant under translation, i.e., k{x-\-E) = X(E) for E in E and x in ...

Every such group has a non-negative countably additive

**measure**which isdefined on the Borel sets E, finite on compact sets, positive or infinite on open

sets, invariant under translation, i.e., k{x-\-E) = X(E) for E in E and x in ...

Page 1152

The existence of an invariant

countability was first shown by Haar [1], and the question ... Other results

concerning

Ulam [1].

The existence of an invariant

**measure**on a group satisfying the second axiom ofcountability was first shown by Haar [1], and the question ... Other results

concerning

**measures**invariant under transformations are found in Oxtoby andUlam [1].

Page 1153

Since the

integration as developed in Chapter III may be used as a basis for the theory

developed in Sections 3 — 4. In particular we should notice that the product

group Rx R is ...

Since the

**measure**space (R, E, A) is a a-finite**measure**space the theory ofintegration as developed in Chapter III may be used as a basis for the theory

developed in Sections 3 — 4. In particular we should notice that the product

group Rx R is ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

Copyright | |

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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