## Linear Operators: Spectral operators |

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Page 860

A

a

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

A

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

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

Page 868

Commutative

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

a

Commutative

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

a

**B**-**algebra**if \} is closed (1.13). It is readily seen that every ideal & in & which ...Page 875

The algebra B(S)) of all bounded linear operators in Hilbert space X) in which the

operation * of involution is defined by equation (i) is a

objective in this section is to characterize commutative B*-algebras. It will be

shown ...

The algebra B(S)) of all bounded linear operators in Hilbert space X) in which the

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 be

shown ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero