## Linear Operators: Spectral operators |

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Page 1218

The following lemma is, in the case of scalar functions, a well known theorem of

Lusin. 17 LEMMA. Let u be a finite positive regular measure on the Borel sets of a

topological space R. Then, for every B-space valued

The following lemma is, in the case of scalar functions, a well known theorem of

Lusin. 17 LEMMA. Let u be a finite positive regular measure on the Borel sets of a

topological space R. Then, for every B-space valued

**u**-**measurable**function f on ...Page 1221

Thus q, is the intersection of a sequence of measurable sets, and it follows that o,

is

the theorem, suppose that the functions Wi(-, 2), ..., W,(-, 2) are not linearly ...

Thus q, is the intersection of a sequence of measurable sets, and it follows that o,

is

**u**-**measurable**, completing the proof of statement (i). To complete the proof ofthe theorem, suppose that the functions Wi(-, 2), ..., W,(-, 2) are not linearly ...

Page 1341

Let {u,} be a positive matria measure whose elements are continuous with

respect to a positive g-finite measure u. If {m,} is the matria of densities of u, with

respect to u, then there exist nonnegative

Let {u,} be a positive matria measure whose elements are continuous with

respect to a positive g-finite measure u. If {m,} is the matria of densities of u, with

respect to u, then there exist nonnegative

**u**-**measurable**functions p, , i = 1,..., n, ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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