## Linear Operators: Spectral operators |

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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 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

...

Page 1624

Let us indicate briefly how the

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

K1: 6°K., & K, r 6t? Gs” — q(t)K1(s, t) with the boundary conditions K1(t, 0) = 0, ...

Let us indicate briefly how the

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

K1: 6°K., & K, r 6t? Gs” — q(t)K1(s, t) with the boundary conditions K1(t, 0) = 0, ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero