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

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Page 1241

given a > 0 there is an integer N ...

and thus limn - 40012,1 + = 0. Since || xml + -lynl + I S \ Xm - ynl + = 12,1+ , we

see ...

**Consequently**there is a number M such that 12m + SM , m = 1 , 2 , .... Moreover ,given a > 0 there is an integer N ...

**Consequently**, limno sup ( 12,1+ ) 2 = Me ,and thus limn - 40012,1 + = 0. Since || xml + -lynl + I S \ Xm - ynl + = 12,1+ , we

see ...

Page 1383

With boundary conditions A , the eigenvalues are

from the equation sin vā = 0 .

numbers of the form ( na ) , n 2 1 ; in Case C , the numbers { ( n + ] ) a } " , n 2 0.

With boundary conditions A , the eigenvalues are

**consequently**to be determinedfrom the equation sin vā = 0 .

**Consequently**, in Case A , the eigenvalues are thenumbers of the form ( na ) , n 2 1 ; in Case C , the numbers { ( n + ] ) a } " , n 2 0.

Page 1387

with the kernel s < t , Il > 0 t < s , Il > 0 sin Văs ( cos Vīt + i sin Vāt ) va sin Vāt ( cos

văs + i sin Văs ) va sin Văs ( cos Vā - i sin Vīt ) va sin Vāt ( cos Vīs- i sin Vās ) va ...

**Consequently**, by Theorem 3.16 , the resolvent R ( 2 ; T ) is an integral operatorwith the kernel s < t , Il > 0 t < s , Il > 0 sin Văs ( cos Vīt + i sin Vāt ) va sin Vāt ( cos

văs + i sin Văs ) va sin Văs ( cos Vā - i sin Vīt ) va sin Vāt ( cos Vīs- i sin Vās ) va ...

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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