## Linear Operators: Spectral theory |

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Page 860

A B-

a fi-

(olx)* = ax*, (#*)* = x. All of the examples mentioned above, with the exception ...

A B-

**algebra**X is commutative in case xy = yx for all x and y in X. An involution ina fi-

**algebra**X is a mapping x -> a;* of X into itself with the properties = (iry)* = y*x*(olx)* = ax*, (#*)* = x. All of the examples mentioned above, with the exception ...

Page 868

Commutative fi-Algebras In case X is a commutative B-

two-sided and the quotient

a B-

Commutative fi-Algebras In case X is a commutative B-

**algebra**every ideal 0 istwo-sided and the quotient

**algebra**X/Q is again a commutative**algebra**. It will bea B-

**algebra**if $ is closed (1.13). It is readily seen that every ideal 3 in X which ...Page 979

One of these algebras, namely the

met before. For convenience, its definition and some of its properties will be

restated here. For every / in L^R) the convolution (/ * gm = \Ri{x-y)g{y)dy, g e L2(R

), ...

One of these algebras, namely the

**algebra**21 of the preceding section, we havemet before. For convenience, its definition and some of its properties will be

restated here. For every / in L^R) the convolution (/ * gm = \Ri{x-y)g{y)dy, g e L2(R

), ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive 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