## Linear Operators: Spectral theory |

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Page 891

scalar

chapter we shall only integrate bounded

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

of a set ...

Page 1178

Then 36 maps scalar-

from Plancherel's theorem that 3% is a bounded mapping of the space L2 of

scalar-

Then 36 maps scalar-

**valued functions**into functions with values in l, . It is plainfrom Plancherel's theorem that 3% is a bounded mapping of the space L2 of

scalar-

**valued functions**into the space La(l,) of square-integrable vector-valued ...Page 1179

It is clear from (63) and (61) that %.4 maps G into the scalar-

whose Fourier transform is defined by (64) f($) = g,(::), 2" < |S| < 2"+". This proves

the right-hand inequality in (54). Q.E.D. 26 CoRollARY. Theorem 25 remains

valid ...

It is clear from (63) and (61) that %.4 maps G into the scalar-

**valued function**fwhose Fourier transform is defined by (64) f($) = g,(::), 2" < |S| < 2"+". This proves

the right-hand inequality in (54). Q.E.D. 26 CoRollARY. Theorem 25 remains

valid ...

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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