## Linear Operators: Spectral theory |

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Page 861

If To

Tile , then az = TÖ ' z for every z e X . Also xa = T a = e = Tr ? ( T & C ) = TĚ ( ex ) =

( Tile ) x = ax . Thus x - 1

If To

**exists**in B ( X ) , then Tx [ ( Tz + y ) ] = yz , ( To ' y ) z = T ? ( y2 ) , and if a =Tile , then az = TÖ ' z for every z e X . Also xa = T a = e = Tr ? ( T & C ) = TĚ ( ex ) =

( Tile ) x = ax . Thus x - 1

**exists**and Tol % = x - 1 % . 2 DEFINITION . An element ...Page 1057

2 ( 7 ) = lim Weius du } F ) ( u ) , 1 ; Jen ly " ^ En lyn Xily provided only that the limit

in the braces in this last equation

present lemma , it suffices to show that ( 3 ) O ( u ) = P | Ply ) eivu dy = lim 2 ( y )

ciou ...

2 ( 7 ) = lim Weius du } F ) ( u ) , 1 ; Jen ly " ^ En lyn Xily provided only that the limit

in the braces in this last equation

**exists**. Thus , to complete the proof of thepresent lemma , it suffices to show that ( 3 ) O ( u ) = P | Ply ) eivu dy = lim 2 ( y )

ciou ...

Page 1261

23 If an operator T has a closed linear extension there

linear extension T such that if T , is any closed linear extension of T then T CT . T

is called the closure of T . ( a ) There

23 If an operator T has a closed linear extension there

**exists**a unique closedlinear extension T such that if T , is any closed linear extension of T then T CT . T

is called the closure of T . ( a ) There

**exists**a densely defined operator with no ...### What people are saying - Write a review

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 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 operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently consider 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 multiplicity neighborhood norm obtained partial positive preceding present problem 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 vanishes vector zero