## Linear Operators: Spectral theory |

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Page 950

... union of countably many compact sets. Every such group has a non-negative

countably additive

compact sets, positive or infinite on open sets, invariant under translation, i.e., A(a

+ E) ...

... union of countably many compact sets. Every such group has a non-negative

countably additive

**measure**which is defined on the Borel sets 2, finite oncompact sets, positive or infinite on open sets, invariant under translation, i.e., A(a

+ E) ...

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

R × R is ...

Since the

**measure**space (R, 2, 2) is a g-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

R × R is ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers 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 f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero