## Linear Operators: Spectral theory |

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Page 1100

matrix with the same eigenvalues as A, the statement (3) follows from the

inequality tr(VD, V"D, V") = |D., D.,. ... Since the linear functional tris continuous on

C1, and since, as we have observed in the paragraph

→ AB ...

matrix with the same eigenvalues as A, the statement (3) follows from the

inequality tr(VD, V"D, V") = |D., D.,. ... Since the linear functional tris continuous on

C1, and since, as we have observed in the paragraph

**following Lemma**9, [A, B)→ AB ...

Page 1102

... in a Hilbert space of dimension d > 00. The formula e" 4 = det(A) is valid for

finite-dimensional matrices, det(A) denoting the determinant of A. Since the

determinant of A is the product of its eigenvalues, it

we have - ...

... in a Hilbert space of dimension d > 00. The formula e" 4 = det(A) is valid for

finite-dimensional matrices, det(A) denoting the determinant of A. Since the

determinant of A is the product of its eigenvalues, it

**follows**by**Lemma**6(a) thatwe have - ...

Page 1696

By Lemma 14 there is a sequence {F,} of elements of D(I), each of which has a

carrier which is a compact subset C, of I, ... it

3.12 that there is a unique extension G of F to a distribution in D,(D) such that the

...

By Lemma 14 there is a sequence {F,} of elements of D(I), each of which has a

carrier which is a compact subset C, of I, ... it

**follows from Lemmas**13, 3.43 and3.12 that there is a unique extension G of F to a distribution in D,(D) such that the

...

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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