## Linear Operators: Spectral theory |

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Page 1347

This procedure has the

of the complex variable A; but it has drawbacks which, though less

nevertheless decisive. Suppose, for example, that we study the self adjoint ...

This procedure has the

**evident**advantage that it makes a,(., A) an entire functionof the complex variable A; but it has drawbacks which, though less

**evident**, arenevertheless decisive. Suppose, for example, that we study the self adjoint ...

Page 1695

It is easily seen that we may find a sequence {C} of compact sets with C, CI, j = 1,.

.., n, and such that U-1C, = I. Using Lemma 2.1, let the function p, in Co(I) be such

that 0 < y,(r) = 1 for all w, and such that p,(a) = 1 for a in C,. Then it is

It is easily seen that we may find a sequence {C} of compact sets with C, CI, j = 1,.

.., n, and such that U-1C, = I. Using Lemma 2.1, let the function p, in Co(I) be such

that 0 < y,(r) = 1 for all w, and such that p,(a) = 1 for a in C,. Then it is

**evident**...Page 1759

Since g satisfies the partial differential equation 6.g(r; s) =Xa,(r; s)6.4(r; s)+b(r; s)g

(r; s), [ar, s] e C1, j=1 it is

equation that any derivative 6'g of g of order at most k can be expressed as a

linear ...

Since g satisfies the partial differential equation 6.g(r; s) =Xa,(r; s)6.4(r; s)+b(r; s)g

(r; s), [ar, s] e C1, j=1 it is

**evident**on repeated partial differentiation of thisequation that any derivative 6'g of g of order at most k can be expressed as a

linear ...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

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 satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero