## Linear Operators: Spectral operators |

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Page 868

This mapping a -> x(JR) of 3 into the field p of

homomorphism. Since la (JR) s r. this homomorphism is continuous. 2 LEMMA.

Let u be a non-zero homomorphism of the commutative B-algebra & onto the field

of ...

This mapping a -> x(JR) of 3 into the field p of

**complex numbers**is clearly ahomomorphism. Since la (JR) s r. this homomorphism is continuous. 2 LEMMA.

Let u be a non-zero homomorphism of the commutative B-algebra & onto the field

of ...

Page 872

The number r(A) is clearly independent of the particular sequence {P,} used to

define it. For a fixed Zoe G the map a -- a (20) is a homomorphism of 3 into the

field of

Jo) ...

The number r(A) is clearly independent of the particular sequence {P,} used to

define it. For a fixed Zoe G the map a -- a (20) is a homomorphism of 3 into the

field of

**complex numbers**. Thus by Lemma 2 there is a maximal ideal Joo with a (Jo) ...

Page 1157

Then a

function g which is analytic in a neighborhood of t and is such that g(z) = f(z) for

all z in this meighborhood for which |z| # 1. Making use of this theorem and an ...

Then a

**complex number**t of modulus 1 is outside a (d) if and only if there exists afunction g which is analytic in a neighborhood of t and is such that g(z) = f(z) for

all z in this meighborhood for which |z| # 1. Making use of this theorem and an ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

52 other sections not shown

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### 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