## Linear Operators, Part 2 |

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Page 891

scalar

chapter we shall only integrate bounded

discussion of the integral will be restricted to that case. Let Z be a field of subsets

of a set ...

scalar

**function**f with respect to the operator**valued**set**function**E. In the presentchapter we shall only integrate bounded

**functions**f and so the followingdiscussion of the integral will be restricted to that case. Let Z be a field of subsets

of a set ...

Page 1178

Let the transformation .1" be defined by the equation <59) <3'/><s) = K<s>f<e>. or

equivalently <60) (1'l)(w) = ffj_f'K<w—y>r<y>dyThen .1' maps scalar-

...

Let the transformation .1" be defined by the equation <59) <3'/><s) = K<s>f<e>. or

equivalently <60) (1'l)(w) = ffj_f'K<w—y>r<y>dyThen .1' maps scalar-

**valued****functions**into functions with values in Z2. It is plain from Plancherel's theorem that...

Page 1179

+00 <68) 1'G<s> = >;°K,<s>§,(s>By Plancherel's theorem, Y is a bounded

mapping of L,(l,) into the space of scalar-

19 and Corollary 17, .2' is a bounded mapping of L,(l,) into L'. It is clear from (63)

and ...

+00 <68) 1'G<s> = >;°K,<s>§,(s>By Plancherel's theorem, Y is a bounded

mapping of L,(l,) into the space of scalar-

**valued functions**I, . Thus, by Corollary19 and Corollary 17, .2' is a bounded mapping of L,(l,) into L'. It is clear from (63)

and ...

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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