## Linear Operators: Spectral theory |

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Page 1527

To deal with the confluent hypergeometric

some of the theory of

Lj = (d/dz)*f-i-p(z)(d/dz)f-i-q(z)f = 0 to have an irregular singularity of order (or type

) ...

To deal with the confluent hypergeometric

**equation**we must consequently usesome of the theory of

**equations**with ... It is most convenient to take the**equation**Lj = (d/dz)*f-i-p(z)(d/dz)f-i-q(z)f = 0 to have an irregular singularity of order (or type

) ...

Page 1556

What is the relationship between 0(t) and the number of zeros of a solution of the

above

operator t has two boundary values at infinity, then N(t) lim = OO, t—-oo 2 where ...

What is the relationship between 0(t) and the number of zeros of a solution of the

above

**equation**? G14 Use the result of the preceding exercise to show that if theoperator t has two boundary values at infinity, then N(t) lim = OO, t—-oo 2 where ...

Page 1685

Since p > n the number q defined by the

< q ~ n/(n-1) and so h, is in L,(E"). Since F, is in L,(E') it follows from Hölder's

inequality that F,(-)h,(:) is in L1(E"). We have, by an elementary change of

variable, ...

Since p > n the number q defined by the

**equation**p-o-Ho-1 = 1 is in the interval 1< q ~ n/(n-1) and so h, is in L,(E"). Since F, is in L,(E') it follows from Hölder's

inequality that F,(-)h,(:) is in L1(E"). We have, by an elementary change of

variable, ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero