## Linear Operators: Spectral theory |

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Page 1000

A similar argument shows that in ( z ) ▻ fy ( z )

A similar argument shows that in ( z ) ▻ fy ( z )

**uniformly**on each compact subset on the half - plane I ( 2 ) > 0. If { In } were known to be**uniformly**convergent in a neighborhood of U , the analyticity of its limit fy would be clear ...Page 1001

It is clear that in converges

It is clear that in converges

**uniformly**on any portion of Q whose closure contains neither a nor b . Let M be a bound for the sequence Yn so that M \ n ( a + is ) ] = M le - iax les -138 < 0 . 1s S $ * - * dx = In the same way it is ...Page 1108

< Σ 12 il If 2 ; ( m ) is a parameter family of sequences such that ( i ) 2 ; ( m ) → h ; as m → 00

< Σ 12 il If 2 ; ( m ) is a parameter family of sequences such that ( i ) 2 ; ( m ) → h ; as m → 00

**uniformly**in i , ( ii ) lim - Lier 12 : ( m ) | * = 0 )**uniformly**in m , , it follows that ( iii ) Lien 12 ; ( m ) / * is bounded ...### 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