## Linear Operators, Part 2 |

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Page 1045

The convolution

operators in Lp(E"), and conditions will be given under which it may be asserted

that the linear mapping T,' : f -> k =1: f is a bounded operator in L,(E"). If j'8..]k(y)[

dy ...

The convolution

**integrals**<1) um/><=») = f,,k<w-y)/(my will be considered asoperators in Lp(E"), and conditions will be given under which it may be asserted

that the linear mapping T,' : f -> k =1: f is a bounded operator in L,(E"). If j'8..]k(y)[

dy ...

Page 1046

an

Cauchy principal value as +OOe(3y -6 00 etzw 1 :1 +H—e _ E-+0 _ 5 z 00 '2 on -

0° el'3Y__e_'-'7' = lim i— da: £—r0 5 'Z co . _ _ SID ly = lim 21]' dc: €—>0 5 w _ _

...

an

**integral**studied by Hilbert. The**integral**(2) may be interpreted in terms of aCauchy principal value as +OOe(3y -6 00 etzw 1 :1 +H—e _ E-+0 _ 5 z 00 '2 on -

0° el'3Y__e_'-'7' = lim i— da: £—r0 5 'Z co . _ _ SID ly = lim 21]' dc: €—>0 5 w _ _

...

Page 1047

If we tried to take |.2:|'1 as the convolution kernel, i.e., if we considered the

multi-dimensional case the convolution

form ...

If we tried to take |.2:|'1 as the convolution kernel, i.e., if we considered the

**integral**J'+°° f(¢) (Lt -00 instead of (8), all our considerations would fail. In themulti-dimensional case the convolution

**integrals**<4) /+0” Q<""") /(1/)dy -00 of theform ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients 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 hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma 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