## Linear Operators: Spectral theory |

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Page 1130

Conversely, if Ku is any family of

Hilbert-Schmidt operator in §o 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 §o 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 K(a,b)f(b),(d)",(da) A A <|s,[s, Ka, b)" from Theorem III.2.20, Theorem III.

11.17, and Schwarz' inequality; thus the integral on the right of (2) defines a

bounded operator K. It is plain from the definition of the

"[s, ...

We have K(a,b)f(b),(d)",(da) A A <|s,[s, Ka, b)" from Theorem III.2.20, Theorem III.

11.17, and Schwarz' inequality; thus the integral on the right of (2) defines a

bounded operator K. It is plain from the definition of the

**kernel**K that 1/2 r(da)|r(a)"[s, ...

Page 1624

Let us indicate briefly how the

known. A formal differentiation gives the following partial differential equation for

K1: 6°K, 8°K. 9t? 8s? = q(t)K1(s,t) with the boundary conditions K1(t, 0) = 0, — (1)

...

Let us indicate briefly how the

**kernel**Kı is obtained once the functions f(t, 2) areknown. A formal differentiation gives the following partial differential equation for

K1: 6°K, 8°K. 9t? 8s? = q(t)K1(s,t) with the boundary conditions K1(t, 0) = 0, — (1)

...

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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