## Linear Operators: Spectral theory |

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Page 876

**Hence**( 1 + N ) 2 = ly + Niel2 = | ( y + Nie ) ( y + Nie ) * 1 = | ( y + Nie ) ( y - Nie ) ) = | y2 + Nael = \ y2I + N2 . Since this inequality must hold for all real N , a contradiction is obtained by placing N \ 94 ) .**Hence**B + d = 0 ...Page 1027

**Hence**a belongs to the spectrum of ET . Conversely , suppose that a non - zero scalar 2 belongs to the spectrum of ET . Then , for some non - zero x in EH , we have ETx = 2x . Then Tx = hæ + y , where y belongs to the subspace ( I - E ) ...Page 1227

**Hence**T * x = ix , or x e Dt .**Hence**D. is closed . Similarly , D_ is closed . Since D7 and D are clearly linear subspaces of D ( T * ) , it remains to show that the spaces D ( T ) , D. , and D are mutually orthogonal , and that their ...### What people are saying - Write a review

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero