## Linear Operators, Part 2 |

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Page 1559

As usual , let N ( t ) be the number of zeros in the interval [ 0 , t ) of a solution of

the equation of = 0 , and

< € < 2 . ( a ) Show that S ' Q ( s ) - 2dN ( 8 ) < Sie ( s ) - 1ds + KS ; 2 - 1 - 4dQ ( 8 )

.

As usual , let N ( t ) be the number of zeros in the interval [ 0 , t ) of a solution of

the equation of = 0 , and

**suppose**that N ( t ) s * Q ( s ) ds + KQ ( € ) 2 - € where 0< € < 2 . ( a ) Show that S ' Q ( s ) - 2dN ( 8 ) < Sie ( s ) - 1ds + KS ; 2 - 1 - 4dQ ( 8 )

.

Page 1563

G41

belongs to the essential spectrum of T . ( a ) Let { { n } be a sequence in D ( T . ( t )

) such that fel = 1 , 1t1n1 → 0 , and such that in vanishes in the interval [ 0 . n ) .

G41

**Suppose**that the function q is bounded below .**Suppose**that the originbelongs to the essential spectrum of T . ( a ) Let { { n } be a sequence in D ( T . ( t )

) such that fel = 1 , 1t1n1 → 0 , and such that in vanishes in the interval [ 0 . n ) .

Page 1597

( 18 ) In the interval [ 0 , 00 ) ,

( t ) ) ( b ) = 0 , t19 ( t ) / 3 poo ( q ' ( t ) ) 2 ( c ) JM 19 ( t ) / 5 / 2 at < 0o , for large M .

Then the essential spectrum of t is empty ( Wintner [ 8 ] ) . ( 19 ) In the interval [ a ...

( 18 ) In the interval [ 0 , 00 ) ,

**suppose**that ( a ) lim g ( t ) = - 00 , t + 00 lim sup ( a '( t ) ) ( b ) = 0 , t19 ( t ) / 3 poo ( q ' ( t ) ) 2 ( c ) JM 19 ( t ) / 5 / 2 at < 0o , for large M .

Then the essential spectrum of t is empty ( Wintner [ 8 ] ) . ( 19 ) In the interval [ a ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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