## Linear Operators: Spectral theory |

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Page 1241

given e > 0 there is an integer N such that if m, n > N, then |zn –z, t < e. Thus (|z,t)*

< (z, , z,,)"|+|(z, , z, -z,,)t| < |(z, , z,)"|+Me. Since (z, , zn.)"| = |(-, . Tom) is 2, Tz,n], lim,

...

**Consequently**there is a number M such that |z, t < M, m = 1, 2, .... Moreover,given e > 0 there is an integer N such that if m, n > N, then |zn –z, t < e. Thus (|z,t)*

< (z, , z,,)"|+|(z, , z, -z,,)t| < |(z, , z,)"|+Me. Since (z, , zn.)"| = |(-, . Tom) is 2, Tz,n], lim,

...

Page 1383

With boundary conditions A, the eigenvalues are

from the equation sin V2 = 0.

numbers of the form (nar)”, n > 1; in Case C, the numbers {(n+})*}”, n > 0.

With boundary conditions A, the eigenvalues are

**consequently**to be determinedfrom the equation sin V2 = 0.

**Consequently**, in Case A, the eigenvalues A are thenumbers of the form (nar)”, n > 1; in Case C, the numbers {(n+})*}”, n > 0.

Page 1387

the kernel sin Västeos votisin vo), , , , , , o, v2. in Vätscos V2s +i sin Vas Sln (cos *

H Sln ). t 3 s, JoA S- 0, v2 sin Visscos Vâl-i sin Vät sin VAs (cos - 2. Sin ). 8 -3 t ...

**Consequently**, by Theorem 3.16, the resolvent R(A; T) is an integral operator withthe kernel sin Västeos votisin vo), , , , , , o, v2. in Vätscos V2s +i sin Vas Sln (cos *

H Sln ). t 3 s, JoA S- 0, v2 sin Visscos Vâl-i sin Vät sin VAs (cos - 2. Sin ). 8 -3 t ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### Other editions - View all

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

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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