## Linear Operators, Part 2 |

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Page 868

I / 3 is a closed ideal in the commutative B-algebra I then the quotient algebra X/3

is isometrically isomorphic to the field 0/

maximal. PROOF. If S is not maximal it is properly contained in an ideal and so I/3

...

I / 3 is a closed ideal in the commutative B-algebra I then the quotient algebra X/3

is isometrically isomorphic to the field 0/

**complex numbers**if and only if 8 ismaximal. PROOF. If S is not maximal it is properly contained in an ideal and so I/3

...

Page 1156

Suppose that ¢ = {an}, -00 < n < +00. is a bounded sequence of

ml Then a

...

Suppose that ¢ = {an}, -00 < n < +00. is a bounded sequence of

**complex**:**numbers**. Let f be the function of the ... 2 a_,1z", |z| < 1. n=0 f(Z) = F F C F C F C “ml Then a

**complex number**t of modulus 1 is outside 1 l 56 XI. MISCELLANEOUS...

Page 1157

Then a

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

all z in this neighborhood for which |z| aé 1. Making use of this theorem and an ...

Then a

**complex number**t of modulus 1 is outside a(¢f>) if and only if there existsa function g which is analytic in a neighborhood of t and is such that g(z) = f(z) for

all z in this neighborhood for which |z| aé 1. Making use of this theorem and an ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic B-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients 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 hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero