## Linear Operators: Spectral theory |

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Page 891

scalar function / with respect to the operator valued set function E. In the present

chapter we shall only integrate bounded functions / and so the following ... We

shall use the term E-

scalar function / with respect to the operator valued set function E. In the present

chapter we shall only integrate bounded functions / and so the following ... We

shall use the term E-

**simple function**for any function on S having the form n ...Page 892

This shows that if the sequence {/„} of Z-simpIe functions converges in B(S, E) to /

then the sequence {Js /•(*)-®(^*)} of ... Thus for / in B(S, E) the equation \s1(s)g(s)

E{ds) = [jsf(s)E(ds)~] [jag{»)E(d>)] holds for all 2"-

This shows that if the sequence {/„} of Z-simpIe functions converges in B(S, E) to /

then the sequence {Js /•(*)-®(^*)} of ... Thus for / in B(S, E) the equation \s1(s)g(s)

E{ds) = [jsf(s)E(ds)~] [jag{»)E(d>)] holds for all 2"-

**simple functions**g, and hence, ...Page 1218

The following lemma is, in the case of scalar functions, a well known theorem of

Lusin. 17 Lemma. Let ju be a ... Since fi(R) < oo every //-measurable function / is

the limit in //-measure of a sequence {/„} of //-

...

The following lemma is, in the case of scalar functions, a well known theorem of

Lusin. 17 Lemma. Let ju be a ... Since fi(R) < oo every //-measurable function / is

the limit in //-measure of a sequence {/„} of //-

**simple functions**. In view of Corollary...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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### Common terms and phrases

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