## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 1027

Hence a

scalar 2

have ETx = 2x . Then Tx = x + y , where y

...

Hence a

**belongs**to the spectrum of ET . Conversely , suppose that a non - zeroscalar 2

**belongs**to the spectrum of ET . Then , for some non - zero æ in EH , wehave ETx = 2x . Then Tx = x + y , where y

**belongs**to the subspace ( I - E ) H , and...

Page 1116

Then plainly | B9 : 12 < ( 7/7/2 ) 2 < 0 , so that , by Definition 6.1 , B

Hilbert - Schmidt class Cz . If we let Aqi = yi - p / 2 Pi , then A is plainly self adjoint

and A

Then plainly | B9 : 12 < ( 7/7/2 ) 2 < 0 , so that , by Definition 6.1 , B

**belongs**to theHilbert - Schmidt class Cz . If we let Aqi = yi - p / 2 Pi , then A is plainly self adjoint

and A

**belongs**to the class Cr , where r ( 1 – p / 2 ) = p , i.e. , r = p ( 1 - p / 2 ) -1 .Page 1602

Then the point 2

14 ] ) . ( 48 ) Suppose that the function q is bounded below , and let | be a real

solution of the equation ( 1-1 ) = 0 on ( 0 , 0 ) which is not square - integrable but

...

Then the point 2

**belongs**to the essential spectrum of 7 ( Hartman and Wintner [14 ] ) . ( 48 ) Suppose that the function q is bounded below , and let | be a real

solution of the equation ( 1-1 ) = 0 on ( 0 , 0 ) which is not square - integrable but

...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero