## Linear Operators: Spectral theory |

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Page 868

|S defined as the set of all 3 = a +3 with a in S. Conversely every ideal in 3/3 is of

this form. 1 THEOREM. If 3 is a closed ideal in the commutative B-algebra 3. then

the quotient algebra 3./S is isometrically isomorphic to

|S defined as the set of all 3 = a +3 with a in S. Conversely every ideal in 3/3 is of

this form. 1 THEOREM. If 3 is a closed ideal in the commutative B-algebra 3. then

the quotient algebra 3./S is isometrically isomorphic to

**the field**of compler ...Page 891

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

chapter we shall only integrate bounded functions f and so the following

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

of a set ...

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

chapter we shall only integrate bounded functions f and so the following

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

**a field**of subsetsof a set ...

Page 932

114], and similar results for semigroups of contractions are found in Sz.-Nagy [8,

10] (see also Cooper [3]). Sz.-Nagy's original proof depended on the following

theorem due to Neumark [3]. THEOREM. Let S be an abstract set and 2

114], and similar results for semigroups of contractions are found in Sz.-Nagy [8,

10] (see also Cooper [3]). Sz.-Nagy's original proof depended on the following

theorem due to Neumark [3]. THEOREM. Let S be an abstract set and 2

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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