## Linear Operators: Spectral theory |

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Page 919

The second assertion is evident from the definition of U. Q.E.D. In connection with the following theorem it should be recalled that two operators T , and T , in Hilbert space H are said to be unitarily

The second assertion is evident from the definition of U. Q.E.D. In connection with the following theorem it should be recalled that two operators T , and T , in Hilbert space H are said to be unitarily

**equivalent**if they are related by ...Page 920

n unitarily

n unitarily

**equivalent**, i.e. , that Ť VTV - 1 where Vis unitary . Under this assumption it will be shown that there is an ordered representation of H onto 2n = , Lylēn ; j ) relative to T. It will follow from Theorem 10 that U and ...Page 1217

Two ordered representations U and Ū of H relative to T and † respectively , with measures u and ù , and multiplicity sets { en } and { en } will be called

Two ordered representations U and Ū of H relative to T and † respectively , with measures u and ù , and multiplicity sets { en } and { en } will be called

**equivalent**if u = û and ule , 4 ? n ) = 0 ) = üle , 4ểm ) for n = 1 , 2 , .### What people are saying - Write a review

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### Contents

8 | 876 |

859 | 885 |

extensive presentation of applications of the spectral theorem | 911 |

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additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero