## Linear Operators: Spectral operators |

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Page 891

scalar

chapter we shall only integrate bounded

discussion of the integral will be restricted to that case. Let X be a field of subsets

of a set ...

scalar

**function f**with respect to the operator valued set function E. In the presentchapter we shall only integrate bounded

**functions f**and so the followingdiscussion of the integral will be restricted to that case. Let X be a field of subsets

of a set ...

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which are well-known in the case

integration is with respect to Haar measure, as is generally the case, we write da

instead

some ...

which are well-known in the case

**of**Lebesgue measure on the line. Whenintegration is with respect to Haar measure, as is generally the case, we write da

instead

**of**A(dr). Throughout, we denote L,(R, 2, 2) by L,(R). To begin the study,some ...

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if

that the conclusion

+ oc) is replaced by the requirement that

if

**f**is**of**bounded variation in the neighborhood**of**ar. (Hint: Cf. IV.14.17.) 13 Showthat the conclusion

**of**Exercise 12 remains valid if the condition that**f**is in L1(–oo,+ oc) is replaced by the requirement that

**f**be in L2(–00, -i- oc). 14 Show that ...### 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