## Linear Operators: Spectral theory |

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Page 876

Then (y-i-Nie)(2) = y(A) + Ni = i (1 + N), and

< y + Niel” = (y-i-Nie)(y-i-Nie)*| = |(y-i-Nie)(y–Nie)|= y?--Noel -s sy”|-|-N*. Since

this inequality must hold for all real N, a contradiction is obtained by placing N = y

” ...

Then (y-i-Nie)(2) = y(A) + Ni = i (1 + N), and

**hence**1 + NI : y + Niel.**Hence**(1+N)*< y + Niel” = (y-i-Nie)(y-i-Nie)*| = |(y-i-Nie)(y–Nie)|= y?--Noel -s sy”|-|-N*. Since

this inequality must hold for all real N, a contradiction is obtained by placing N = y

” ...

Page 1027

scalar A belongs to the spectrum of ET. Then, for some non-zero a in EX), we

have ETa' = Air. Then Ta' = Aw--y, where y belongs to the subspace (I–E)S), and

...

**Hence**A belongs to the spectrum of ET. Conversely, suppose that a non-zeroscalar A belongs to the spectrum of ET. Then, for some non-zero a in EX), we

have ETa' = Air. Then Ta' = Aw--y, where y belongs to the subspace (I–E)S), and

...

Page 1227

Proof. By Lemma 8(a), Q(T) is closed. Suppose {r,} is a sequence of elements of

Q, converging to re 3 (T"), then {a, , Tor,)} = {a, , ir,J} CI'(T") converges to [æ, ir] = [

a, Tor], since T" is closed.

...

Proof. By Lemma 8(a), Q(T) is closed. Suppose {r,} is a sequence of elements of

Q, converging to re 3 (T"), then {a, , Tor,)} = {a, , ir,J} CI'(T") converges to [æ, ir] = [

a, Tor], since T" is closed.

**Hence**To r = ir, or a e 3), .**Hence**3), is closed. Similarly...

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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 |

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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 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