## Linear Operators: Spectral theory |

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Page 860

A B-

a B-

++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 B-Algebras In case 3 is a commutative B-

two-sided and the quotient

a B-

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 bea B-

**algebra**if Ø is closed (1.13). It is readily seen that every ideal & in 3 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 f in L1(R) the convolution ( . )() = s.sr-y)g(y)ly, ge L.(R), ...

One of these algebras, namely the

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

restated here. For every f in L1(R) the convolution ( . )() = s.sr-y)g(y)ly, ge L.(R), ...

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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