## Linear Operators: Spectral theory |

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Page 1074

8 Show , with the hypotheses and notation of Exercise 6 , that if b is in Lol - 00 , +

00 ) , then sto 16 ( t ) | 2 - - | F ( t ) | pdt < 0 . 9 Let 2 be a real function of a real

variable such that 2 ( • ) F ( : ) is the Fourier

00 ) ...

8 Show , with the hypotheses and notation of Exercise 6 , that if b is in Lol - 00 , +

00 ) , then sto 16 ( t ) | 2 - - | F ( t ) | pdt < 0 . 9 Let 2 be a real function of a real

variable such that 2 ( • ) F ( : ) is the Fourier

**transform**of a function in Lj ( - 00 , +00 ) ...

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15 Show that there exists a function f in L ( - 00 , + 00 ) for which the family of

functions 1 p + A f ( x ) = - T F ( t ) e - itx dt , 21 J - A F denoting the Fourier

not every ...

15 Show that there exists a function f in L ( - 00 , + 00 ) for which the family of

functions 1 p + A f ( x ) = - T F ( t ) e - itx dt , 21 J - A F denoting the Fourier

**transform**of f , fails to satisfy the inequality sup A > OJ ( x ) \ dx < 0 . 16 Show thatnot every ...

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frequently - used device , it is appropriate that we give a brief sketch indicating

how the Cayley

has a self adjoint extension . Let T be a symmetric operator with domain D ( T ) ...

frequently - used device , it is appropriate that we give a brief sketch indicating

how the Cayley

**transform**can be used to determine when a symmetric operatorhas a self adjoint extension . Let T be a symmetric operator with domain D ( T ) ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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additive adjoint operator 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 neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero