## Linear Operators: Spectral operators |

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Page 984

The set of functions f in L1(R) for which f

dense in L1(R). PRoof. It follows from Lemma 3.6 that the set of all functions in L2

(R, 2, u) which

The set of functions f in L1(R) for which f

**vanishes**in a neighborhood of infinity isdense in L1(R). PRoof. It follows from Lemma 3.6 that the set of all functions in L2

(R, 2, u) which

**vanish**outside of compact sets is dense in this space, and from ...Page 1650

If F

parts of this lemma are left to the reader as an exercise. To prove (v), we must

show from our hypothesis that F(q) = 0 if p is in C. (U. I.). Let K be a compact

subset of ...

If F

**vanishes**in each set I, it**vanishes**in U.I.: PRoof. The proofs of the first fourparts of this lemma are left to the reader as an exercise. To prove (v), we must

show from our hypothesis that F(q) = 0 if p is in C. (U. I.). Let K be a compact

subset of ...

Page 1651

hal,” is of of: | - J. o beas: so o let' ... o Fo *** and p

= F. If KCP = 0 and the function p in Co(IU Io)

hal,” is of of: | - J. o beas: so o let' ... o Fo *** and p

**vanishes**outside K, then pkp-p**vanishes**outside a compact subset of I-CF, so that G(p) = F(pkq) = F(q). Thus G|I= F. If KCP = 0 and the function p in Co(IU Io)

**vanishes**outside K, then it is clear ...### 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