## Linear Operators: Spectral theory |

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Page 1187

The inverse of a

only if its domain is

which maps sa, y] into [y, r) then I'(T-1) = A11'(T) which shows that T is

The inverse of a

**closed**operator is**closed**. A bounded operator is**closed**if andonly if its domain is

**closed**. PRoof. If A1 is the isometric automorphism in S) GS S)which maps sa, y] into [y, r) then I'(T-1) = A11'(T) which shows that T is

**closed**if ...Page 1393

Let T be a

such that the range of AI —T is not

and is denoted by o,(T). It is clear that o,(T) Co.(T). If t is a formal differential

operator ...

Let T be a

**closed**operator in Hilbert space. Then the set of complex numbers Asuch that the range of AI —T is not

**closed**is called the essential spectrum of Tand is denoted by o,(T). It is clear that o,(T) Co.(T). If t is a formal differential

operator ...

Page 1394

by what has been shown above )+9t, , is

establish the converse part of the present lemma under the additional hypothesis

that JR is one-dimensional, i.e., that Jo = {x r}. If Tre T?), then T()+9°) = T())), ...

by what has been shown above )+9t, , is

**closed**, it is sufficient for this purpose toestablish the converse part of the present lemma under the additional hypothesis

that JR is one-dimensional, i.e., that Jo = {x r}. If Tre T?), then T()+9°) = T())), ...

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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 |

### Common terms and phrases

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