## Linear Operators: Spectral theory |

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Page 1130

Conversely, if Ku is any family of

Hilbert-Schmidt operator in Soo with norm ... Our first step will be to establish that

every Hilbert-Schmidt operator K in L2(A) is represented by a unique

Conversely, if Ku is any family of

**kernels**satisfying (i), ..., (iv), then (v) defines aHilbert-Schmidt operator in Soo with norm ... Our first step will be to establish that

every Hilbert-Schmidt operator K in L2(A) is represented by a unique

**kernel**K(-, ...Page 1131

We have |s,s, K(a,b)f(b)(d) (da) J. J. K(a. by v(da), (d) o s "[s, f(b),(ab)” from

Theorem III.2.20, Theorem III.11.17, and Schwarz' inequality; thus the integral on

the right of (2) defines a bounded operator R. It is plain from the definition of the

We have |s,s, K(a,b)f(b)(d) (da) J. J. K(a. by v(da), (d) o s "[s, f(b),(ab)” from

Theorem III.2.20, Theorem III.11.17, and Schwarz' inequality; thus the integral on

the right of (2) defines a bounded operator R. It is plain from the definition of the

**kernel**...Page 1590

For a detailed exposition of the problems connected with the calculation of the

Green's

Mohr [1] may be found valuable. Section 4. The work of Hilbert [1] in 1904 already

...

For a detailed exposition of the problems connected with the calculation of the

Green's

**kernel**for a differential operator on a finite interval, the recent paper of E.Mohr [1] may be found valuable. Section 4. The work of Hilbert [1] in 1904 already

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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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 |

### 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