## Linear Operators: Spectral theory |

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Page 860

A

a

++y”, (ry)* = y” (x,v)* = &r", (r”)* = r. All of the examples mentioned above, with the

...

A

**B**-**algebra**3 is commutative in case a y = ya for all a and y in 3. An involution ina

**B**-**algebra**3 is a mapping a -> a “ of 3 into itself with the properties (a +y)* = a++y”, (ry)* = y” (x,v)* = &r", (r”)* = r. All of the examples mentioned above, with the

...

Page 868

Commutative

two-sided and the quotient algebra 3/8 is again a commutative algebra. It will be

a

Commutative

**B**-**Algebras**In case 3 is a commutative**B**-**algebra**every ideal & istwo-sided and the quotient algebra 3/8 is again a commutative algebra. It will be

a

**B**-**algebra**if Ø is closed (1.13). It is readily seen that every ideal & in 3 which ...Page 884

The study of ideal theory in

most of the results given in Section 1 are due. B- and

Section 2 are due to Gelfand [1]. The fundamental Theorem 3.7 was proved by ...

The study of ideal theory in

**B**-**algebra**was inaugurated by Gelfand [1] to whommost of the results given in Section 1 are due. B- and

**B**"-**algebras**. The results ofSection 2 are due to Gelfand [1]. The fundamental Theorem 3.7 was proved by ...

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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