## Linear Operators: Spectral operators |

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Results 1-3 of 82

Page 973

t mapping

e) = Z(t(e)), ee B, where 3 is the family of Borel sets

measure

t mapping

**R**onto all**of R**with the properties [**r**, m) = e^*("), a e**R**, m e**R**and 2:tu(e) = Z(t(e)), ee B, where 3 is the family of Borel sets

**in R**and where 2 is Haarmeasure

**on R**. PRoof. For a fixed m in .40 =**R**, the character [w, m), a e**R**, is a ...Page 1291

Our next task is to prove that if t is regular Ti(

adjoint it will follow that

(cf. Lemma XII.4.3') that every self adjoint extension of

Our next task is to prove that if t is regular Ti(

**r**) = To(t”)*. In case t is formally selfadjoint it will follow that

**To(r**) C T (t) = To(t)*, showing that**To(r**) is symmetric, and(cf. Lemma XII.4.3') that every self adjoint extension of

**To(r**) is a restriction of Ti(**r**).Page 1437

Suppose that a bounded sequence {fi} of elements of Q(

–70)f,} converges but the sequence {fi} has no convergent subsequence. Then,

since

Suppose that a bounded sequence {fi} of elements of Q(

**To(r**)) exists such that {(**r**–70)f,} converges but the sequence {fi} has no convergent subsequence. Then,

since

**To(r**) C Ti(**r**), it follows immediately from the preceding lemma that Zoe ...### 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