## Linear Operators: Spectral theory |

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Page 1297

If A ( t ) = 0 for each function in the domain of T ( T ) which vanishes in a

neighborhood of a , A will be

boundary value at b is defined similarly . By analogy with Definition XII . 4 . 25 an

equation B ...

If A ( t ) = 0 for each function in the domain of T ( T ) which vanishes in a

neighborhood of a , A will be

**called**a boundary value at a . The concept of aboundary value at b is defined similarly . By analogy with Definition XII . 4 . 25 an

equation B ...

Page 1432

In this case , v is

, there is no singularity at all , and zero is

equation . If v = 1 , the singularity of equation [ * ] at zero is

In this case , v is

**called**the order of the singularity of equation [ * ] at zero . If v = 0, there is no singularity at all , and zero is

**called**a regular point of the differentialequation . If v = 1 , the singularity of equation [ * ] at zero is

**called**a regular ...Page 1451

The number c is

the upper ( lower ) bound for T . If ( Tx , x ) 20 for all x in D ( T ) , then T is

nonnegative . 20 DEFINITION . Lett be a formally symmetric formal differential ...

The number c is

**called**a bound for T , and the smallest ( largest ) such c is**called**the upper ( lower ) bound for T . If ( Tx , x ) 20 for all x in D ( T ) , then T is

**called**nonnegative . 20 DEFINITION . Lett be a formally symmetric formal differential ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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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 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