## Linear Operators: Spectral theory |

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Page 1258

Conversely , any such maximal symmetric operator is the infinitesimal generator

of a strongly continuous semi - group of partially

domain H . Hilbert space H may be decomposed into an orthogonal direct sum H

...

Conversely , any such maximal symmetric operator is the infinitesimal generator

of a strongly continuous semi - group of partially

**isometric**operators with initialdomain H . Hilbert space H may be decomposed into an orthogonal direct sum H

...

Page 1272

9 , and that an

, if dd = 0 ) = d _ . Also it is clear that a closed

extension if and only if there is an

...

9 , and that an

**isometric**operator V is unitary if and only if D ( V ) H = R ( V ) , i . e ., if dd = 0 ) = d _ . Also it is clear that a closed

**isometric**operator has a unitaryextension if and only if there is an

**isometric**mapping of Dt onto D _ , and that this...

Page 1373

of L ( 1 , { Pi } ) into L2 ( 1 , { i } ) and an

into L2 ( 1 , Ais } ) . Since { ajj ( a ) } and { bis ( a ) } are inverse matrices , it follows

readily that AB = BA = I . Thus , A and B are

of L ( 1 , { Pi } ) into L2 ( 1 , { i } ) and an

**isometric**isomorphism of L , ( 4 , { Wix } )into L2 ( 1 , Ais } ) . Since { ajj ( a ) } and { bis ( a ) } are inverse matrices , it follows

readily that AB = BA = I . Thus , A and B are

**isometric**isomorphisms onto all of ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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