## Linear Operators: Spectral theory |

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Page 1130

Conversely, if Ktj is any family of

Hilbert-Schmidt operator in §0 with norm ...

that (1) 11*11* = jAjA \K(a, b)\H(aa)v(db) < oo and (2) (Kf)(a)= fAK(a,b)f(b)v(db).

Conversely, if Ktj is any family of

**kernels**satisfying (i), . . ., (iv), then (v) defines aHilbert-Schmidt operator in §0 with norm ...

**kernel**K(', •) e L2(A xA) in such a waythat (1) 11*11* = jAjA \K(a, b)\H(aa)v(db) < oo and (2) (Kf)(a)= fAK(a,b)f(b)v(db).

Page 1131

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

(K<Pi<<Pi) for each i and /. Thus, forming linear combinations and by continuity, (

Kf,g) ...

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**K that (KVs><Pi) =(K<Pi<<Pi) for each i and /. Thus, forming linear combinations and by continuity, (

Kf,g) ...

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 | 860 |

Commutative BAlgebras | 874 |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero