## Linear Operators: Spectral theory |

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Page 1303

Clearly B(f) = 0 for those f which vanish in a

boundary value for t at a. To prove the converse, let B be a boundary value at a.

Choose a function h in C*(I) which is identically equal to one in a

of a ...

Clearly B(f) = 0 for those f which vanish in a

**neighborhood**of a. Thus B is aboundary value for t at a. To prove the converse, let B be a boundary value at a.

Choose a function h in C*(I) which is identically equal to one in a

**neighborhood**of a ...

Page 1403

It is then sufficient to show that each A e A has a

A) e L2(a,c) for ut-almost all A e Ao, since A may then be written as a countable

union of such

It is then sufficient to show that each A e A has a

**neighborhood**Ao such that W,(“,A) e L2(a,c) for ut-almost all A e Ao, since A may then be written as a countable

union of such

**neighborhoods**Ao. We shall show below that for each A e A there ...Page 1678

Let is be a second function in Co(I) such that f(r) = 1 for a in a

. Then pop–9p vanishes in a

op ...

Let is be a second function in Co(I) such that f(r) = 1 for a in a

**neighborhood**of K1. Then pop–9p vanishes in a

**neighborhood**of K n C(F), and vanishes in a**neighborhood**of C(F)—K since p vanishes in the complement of K. Hence opp—op ...

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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