## Linear Operators, Part 2 |

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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 ... function E on a field Z' of subsets of an abstract

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

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

chapter we shall only integrate ... function E on a field Z' of subsets of an abstract

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

**measurable functions**.Page 893

=0US/o>E<ds>] on = [S/'<s>E<o>. and since simple functions are dense in B(S,Z'

), we have LiSf(s)E(ds):|* = fsf(s)E(ds) for all ... 2). is a continuous "'-homomorphic

map of the B"'-algebra B(S,Z) of bounded X-

=0US/o>E<ds>] on = [S/'<s>E<o>. and since simple functions are dense in B(S,Z'

), we have LiSf(s)E(ds):|* = fsf(s)E(ds) for all ... 2). is a continuous "'-homomorphic

map of the B"'-algebra B(S,Z) of bounded X-

**measurable functions**on S into the ...Page 900

If f is X-measurable then fo is a bounded Z-

of the B'-algebra B(S, Z). The algebra EB(S, E) of E-essentially bounded Z'-

classes ...

If f is X-measurable then fo is a bounded Z-

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

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

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### Contents

SPECTRAL THEORY | 858 |

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 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 hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero