## Linear Operators: Spectral theory |

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Page 891

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

chapter we shall only integrate bounded ... E on a field X of subsets of an abstract

set S. The functions we shall integrate are the bounded 2-

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

chapter we shall only integrate bounded ... E on a field X of subsets of an abstract

set S. The functions we shall integrate are the bounded 2-

**measurable functions**.Page 893

[ssor(a)" - $3, e() =s foea) i-1 and since simple functions are dense in B(S, X'), we

have s s f(s)}{ds}]" - s f(s) E(ds) for all f in ... fe B(S. 2), is a continuous *-

homomorphic map of the B"-algebra B(S, X) of bounded 2-

on S into ...

[ssor(a)" - $3, e() =s foea) i-1 and since simple functions are dense in B(S, X'), we

have s s f(s)}{ds}]" - s f(s) E(ds) for all f in ... fe B(S. 2), is a continuous *-

homomorphic map of the B"-algebra B(S, X) of bounded 2-

**measurable functions**on S into ...

Page 900

If f is 2-measurable then fo is a bounded 2-

of the B"-algebra B(S, X). The algebra EB(S, X) of E-essentially bounded 2-

classes ...

If f is 2-measurable then fo is a bounded 2-

**measurable function**, i.e., an elementof the B"-algebra B(S, X). The algebra EB(S, X) of E-essentially bounded 2-

**measurable functions**on S is the B"-algebra whose elements are equivalenceclasses ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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### Common terms and phrases

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