## Linear Operators: Spectral operators |

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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(r

-HE) ...

... 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(r

-HE) ...

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, A) 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 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### Common terms and phrases

adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero