## Linear Operators: Spectral theory |

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Results 1-3 of 88

Page 1401

Define an isometric

Ua', a e O,. Let T be the ... If a e o (T,) then a 1 = 0; therefore the

of 92 into T

Define an isometric

**mapping**U, of or onto 3) as follows: U, a = Ua, a e 3, U, a = —Ua', a e O,. Let T be the ... If a e o (T,) then a 1 = 0; therefore the

**mapping**a -- anof 92 into T

**maps**97 onto all of T. By the two preceding lemmas, 92 is exactly ...Page 1669

Let M : II – I, be a

whenever C is a compact subset of Is; (b) (M(.)), e C*(I), j = 1,..., n2. Then (i) for

each p in C*(I2), p o M will denote the function p in C*(I) defined, for a in II, by the

...

Let M : II – I, be a

**mapping**of II into I, such that (a) M-'C is a compact subset of I,whenever C is a compact subset of Is; (b) (M(.)), e C*(I), j = 1,..., n2. Then (i) for

each p in C*(I2), p o M will denote the function p in C*(I) defined, for a in II, by the

...

Page 1671

=|f(M-(r))w()JG)dr, J denoting the absolute value of the Jacobian determinant of

the

variables in a multiple integral. But then (iii) is evident. Q.E.D. Lemma 47 allows

us to ...

=|f(M-(r))w()JG)dr, J denoting the absolute value of the Jacobian determinant of

the

**mapping**a -> M-"(a); this follows by the standard theorem on change ofvariables in a multiple integral. But then (iii) is evident. Q.E.D. Lemma 47 allows

us to ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 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