## Linear Operators: Spectral theory |

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Page 884

The study of ideal theory in B - algebra was inaugurated by Gelfand [ 1 ] to whom

most of the results

of Section 2 are due to Gelfand [ 1 ] . The fundamental Theorem 3 .

The study of ideal theory in B - algebra was inaugurated by Gelfand [ 1 ] to whom

most of the results

**given**in Section 1 are due . B - and B * - algebras . The resultsof Section 2 are due to Gelfand [ 1 ] . The fundamental Theorem 3 .

Page 909

The proof follows immediately , for since L SM , we have ( Lx , x ) < ( Mx , x ) for

every x in H . Hence the characterization of an , Mn

that hn Mn for all n = 1 , 2 , . . . . 5 . Spectral Representation Let u be a finite

positive ...

The proof follows immediately , for since L SM , we have ( Lx , x ) < ( Mx , x ) for

every x in H . Hence the characterization of an , Mn

**given**in Theorem 3 showsthat hn Mn for all n = 1 , 2 , . . . . 5 . Spectral Representation Let u be a finite

positive ...

Page 1149

likewise

representations , which , if one tries to regard them as representations of the

rotation group itself , turn out to be doublevalued . These representations are the

so ...

likewise

**given**by irreducible sets of tensors . The group RŮ ( n ) has additionalrepresentations , which , if one tries to regard them as representations of the

rotation group itself , turn out to be doublevalued . These representations are the

so ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero