## Linear Operators: Spectral theory |

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Page 1297

If A(f) = 0 for each function in the domain of Ti(r) 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(f) = 0, ...

If A(f) = 0 for each function in the domain of Ti(r) 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(f) = 0, ...

Page 1432

In this case, v is

there is no singularity at all, and zero is

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

singularity; ...

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 y = 1, the singularity of equation [*] at zero is

**called**a regularsingularity; ...

Page 1504

A point zo in the complex plane at which ri and r2 are analytic is

point of the operator. In the neighborhood of a regular point zo, there exists a

unique analytic solution f(z) of the equation Lj = 0 with specified initial values f(zo

), ...

A point zo in the complex plane at which ri and r2 are analytic is

**called**a regularpoint of the operator. In the neighborhood of a regular point zo, there exists a

unique analytic solution f(z) of the equation Lj = 0 with specified initial values f(zo

), ...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

38 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 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