## Linear Operators: Spectral theory |

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Page 942

Now if T9(p = X<p then f g(su~1)(p(u)fi(du) = hp(s), seG, J G is a

seen that J Gg(*u-1)«jp(w< )fi{du) = X<p(st), i.e., every translate <p' of an

eigenfunction ...

Now if T9(p = X<p then f g(su~1)(p(u)fi(du) = hp(s), seG, J G is a

**continuous****function**. By replacing * by st and u by ut and using the fact that fi(Et) = fx(E) it isseen that J Gg(*u-1)«jp(w< )fi{du) = X<p(st), i.e., every translate <p' of an

eigenfunction ...

Page 966

For some choice of / the integral on the right of [*] is not zero and since, by

Lemma 1(d), the integral on the left of [*] is continuous, we conclude that hm

agrees almost everywhere with a

of ...

For some choice of / the integral on the right of [*] is not zero and since, by

Lemma 1(d), the integral on the left of [*] is continuous, we conclude that hm

agrees almost everywhere with a

**continuous function**. By redefining hm on a setof ...

Page 1002

4 If / is a non-negative function in AP, and M(f) = 0 (in the notation of Exercise 2)

then / = 0. 5 A

almost periodic if for each e > 0 there exists a number L(e) such that each circle

in the ...

4 If / is a non-negative function in AP, and M(f) = 0 (in the notation of Exercise 2)

then / = 0. 5 A

**continuous function**/ of two real variables x = (xv x2) is calledalmost periodic if for each e > 0 there exists a number L(e) such that each circle

in the ...

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### Contents

BAlgebras | 859 |

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 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 q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero