## Linear Operators, Part 2 |

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Page 860

A

a

and the class of analytic functions, are commutative

A

**B**-**algebra**I is commutative in case my = ya: for all m and y in I. An involution ina

**B**-**algebra**PE is a mapping rc —> a:* of I ... with the exception of L1(—oo, 00)and the class of analytic functions, are commutative

**B**-**algebras**with involutions.Page 868

Commutative

two-sided and the quotient algebra X/3 is again a commutative algebra. It will be

a

Commutative

**B**-**Algebras**In case I is a commutative**B**-**algebra**every ideal S istwo-sided and the quotient algebra X/3 is again a commutative algebra. It will be

a

**B**-**algebra**ififtis closed (1.13). It is readily seen that every ideal 3 in I which ...Page 875

The

the operation "' of involution is defined by equation (i) is a

objective in this section is to characterize commutative

shown ...

The

**algebra B**(.§)) of all bounded linear operators in Hilbert space Q) in whichthe operation "' of involution is defined by equation (i) is a

**B***-**algebra**. Our chiefobjective in this section is to characterize commutative

**B***-**algebras**. It will beshown ...

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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