## Linear Operators: Spectral theory |

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Page 868

It is seen from Theorem 1 that for each JJ e -o and each as in 3: there is a

3 into the field p of

this ...

It is seen from Theorem 1 that for each JJ e -o and each as in 3: there is a

**complex number**a (JR) such that a H-S) = w(Jo)e-i-J'. This mapping a -> r(S).) of3 into the field p of

**complex numbers**is clearly a homomorphism. Since w()') s athis ...

Page 902

Then, for every Borel set 6 of

is ... A finite number of these neighborhoods cover the set 6, of those

...

Then, for every Borel set 6 of

**complex numbers**, E,(0) = E(f-(6)). PROOF. Since fis ... A finite number of these neighborhoods cover the set 6, of those

**complex****numbers**ż with |2| < M and whose distance from g(T) is at least 1/n. Thus E(f-'(6,))...

Page 1251

In this section we shall show how the spectral theorem for self adjoint operators

may be applied to prove a number of results from the theory ... For if {m,} has such

a representation, and 20, ..., x, is any finite set of

In this section we shall show how the spectral theorem for self adjoint operators

may be applied to prove a number of results from the theory ... For if {m,} has such

a representation, and 20, ..., x, is any finite set of

**complex numbers**n s m;1,2,3, ...### What people are saying - Write a review

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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