## Linear Operators: Spectral theory |

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Page 1454

Q . E . D . 23 LEMMA . If T is a closed symmetric operator in Hilbert space , and T

is bounded below , then ( a ) the essential spectrum of T is a subset of the real

axis which is bounded below ; ( b ) the deficiency indices of T are

.

Q . E . D . 23 LEMMA . If T is a closed symmetric operator in Hilbert space , and T

is bounded below , then ( a ) the essential spectrum of T is a subset of the real

axis which is bounded below ; ( b ) the deficiency indices of T are

**equal**. PROOF.

Page 1539

A6 Let t be a regular formally symmetric formal differential operator on [ 0 , 0 )

with

distance from a to the essential spectrum of t is less than or

A6 Let t be a regular formally symmetric formal differential operator on [ 0 , 0 )

with

**equal**deficiency indices , and let 2 be a real number . Prove that thedistance from a to the essential spectrum of t is less than or

**equal**to K if and ...Page 1735

Let s in CO ( En ) be identically

sphere in En and identically zero outside the sphere of radius 2 in E " . We wish

to show that f | U is in H ( + 1 ) ( UI ) for some neighborhood U of the origin . We

see ...

Let s in CO ( En ) be identically

**equal**to 1 in a neighborhood of the unit closedsphere in En and identically zero outside the sphere of radius 2 in E " . We wish

to show that f | U is in H ( + 1 ) ( UI ) for some neighborhood U of the origin . We

see ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 give 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 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 unit vanishes vector zero