## Linear Operators: Spectral theory |

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Page 1174

is in none of the sets e. Then (43) | g(s)

the similar well-known equation for scalarvalued functions. This concludes the ...

**Let**h be a scalar-valued function and put g(s) = h(s)a? if s is in e, and g(s) = 0 if sis in none of the sets e. Then (43) | g(s)

**f**(s)is = sh9)**f**(s)'ds; hence (42) follows fromthe similar well-known equation for scalarvalued functions. This concludes the ...

Page 1200

accordance with Definition 5. Corollary 7 shows that [*] f,(T) D fi(T). Let e be a

bounded Borel set of reals so that, by Theorem 3, E(e)) C3 (T) and TE(e) = s. AE(

dź).

**Let f**(T) be this operator and**let f**,(T) be the operator corresponding to finaccordance with Definition 5. Corollary 7 shows that [*] f,(T) D fi(T). Let e be a

bounded Borel set of reals so that, by Theorem 3, E(e)) C3 (T) and TE(e) = s. AE(

dź).

Page 1649

Let I be an open set in E", and

I defined by the equation F(p) = F(j), q e Co(I), is called the compler conjugate of

F. 8 LEMMA. Let I and F be as in the preceding definition, and let r be a formal ...

Let I be an open set in E", and

**let F**be a distribution in I. Then the distribution F inI defined by the equation F(p) = F(j), q e Co(I), is called the compler conjugate of

F. 8 LEMMA. Let I and F be as in the preceding definition, and let r be a formal ...

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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