## Linear Operators: Spectral theory |

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Page 890

... be 0 if 6 contains none of the spectrum {A1, ..., A) of T and otherwise let E(6) be

the sum of all the projections E(A,) for which 2, e 6, then the function E is a

resolution of the identity for T and the operational calculus is given by the

(vi) ...

... be 0 if 6 contains none of the spectrum {A1, ..., A) of T and otherwise let E(6) be

the sum of all the projections E(A,) for which 2, e 6, then the function E is a

resolution of the identity for T and the operational calculus is given by the

**formula**(vi) ...

Page 1089

This

... (p, q'a) = 0 Since Top|* = (Tp, Top) = (T* Top, p), we see our lemma to be a

special case of the “minimax

...

This

**formula**may be written (u,(T))* = mi IIlax |Tops”. W1, . . . , Won |Q|=1 (p, q'i) =... (p, q'a) = 0 Since Top|* = (Tp, Top) = (T* Top, p), we see our lemma to be a

special case of the “minimax

**formula**” for the eigenvalues of a compact operator,...

Page 1363

basis for this

projection in the resolution of the identity for T corresponding to (A1, A2) may be

calculated from the resolvent by the

ie; ...

basis for this

**formula**is found in Theorem XII.2.10 which asserts that theprojection in the resolution of the identity for T corresponding to (A1, A2) may be

calculated from the resolvent by the

**formula**1 FAA-8 E((A1, A,))f = lim lim - [R(4—ie; ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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 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 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