## Linear Operators: Spectral theory |

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Page 1236

We shall prove the second statement first . As each B ; is a continuous linear

functional on D ( T * ) vanishing on D ( T ) , it is

closed extension of T. Since the set of boundary conditions is symmetric , it

follows ...

We shall prove the second statement first . As each B ; is a continuous linear

functional on D ( T * ) vanishing on D ( T ) , it is

**clear**from Lemma 5 ( c ) that T is aclosed extension of T. Since the set of boundary conditions is symmetric , it

follows ...

Page 1652

Then , since Fl « ) 2 Fl , for each F in H ( * ) ( I ) , it is

some F in L ( I ) . Similarly , since Fl « 2100F , for each F in H ( * ) ( I ) and each

index J such that Jl Sk , it is

to ...

Then , since Fl « ) 2 Fl , for each F in H ( * ) ( I ) , it is

**clear**that { F , } converges tosome F in L ( I ) . Similarly , since Fl « 2100F , for each F in H ( * ) ( I ) and each

index J such that Jl Sk , it is

**clear**that if ( Jl Sk , the sequence { a'Fn } convergesto ...

Page 1689

Indeed , if { { m } is a Cauchy sequence in L ( 1 ) , it is

a Cauchy sequence in L ( 1 ) for J Sk , so that there exist functions g , gl in L , ( 1 )

such that limm - com - gl , = 0 and limm - c001m - g'lp = 0. It is then

Indeed , if { { m } is a Cauchy sequence in L ( 1 ) , it is

**clear**from ( i ) that { a'im } isa Cauchy sequence in L ( 1 ) for J Sk , so that there exist functions g , gl in L , ( 1 )

such that limm - com - gl , = 0 and limm - c001m - g'lp = 0. It is then

**clear**from ...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

### Common terms and phrases

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