## 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 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero