## Linear Operators: Spectral theory |

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Page 1272

If T is a symmetric operator with dense domain , then it has proper symmetric

extensions provided both of its deficiency

maximal symmetric operator is one which has no proper symmetric extensions ;

hence ...

If T is a symmetric operator with dense domain , then it has proper symmetric

extensions provided both of its deficiency

**indices**are different from zero . Amaximal symmetric operator is one which has no proper symmetric extensions ;

hence ...

Page 1398

Let t be a formally self adjoint formal differential operator defined on an interval I .

If the minimum of the deficiency

equation to = ho has at least k linearly independent solutions in L2 ( I ) . Proof .

Let t be a formally self adjoint formal differential operator defined on an interval I .

If the minimum of the deficiency

**indices**of To ( t ) is k , then for a € 0e ( t ) theequation to = ho has at least k linearly independent solutions in L2 ( I ) . Proof .

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

.

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.

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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