## Linear Operators, Part 2 |

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Page 891

Let Z be a field of subsets of a

algebra of bounded linear operators on the B-space X. It is assumed that E is

Let Z be a field of subsets of a

**set**S and let E be a**function**which maps E into thealgebra of bounded linear operators on the B-space X. It is assumed that E is

**additive**and bounded, i.e., there is a constant K such that for every pair 6, 0' of ...Page 958

U e2l'P(e1 U ea) = lE(e1)+E(e2)l'P(*-'1 U ea) = E(e1l'/'(e1 U ¢al+E(e2l'P(e1 U ea)

= 'P(e1)+'/lies)» so that the vector valued

Therefore, if el n ea = ¢, the

% .3' ...

U e2l'P(e1 U ea) = lE(e1)+E(e2)l'P(*-'1 U ea) = E(e1l'/'(e1 U ¢al+E(e2l'P(e1 U ea)

= 'P(e1)+'/lies)» so that the vector valued

**set function**lp is**additive**on Q0.Therefore, if el n ea = ¢, the

**set function**no satisfies the equation ll ll /5 f'\ I¢\ '€ '€% .3' ...

Page 1899

(See Continuous set function and Set function) space of, additional properties, IV.

15 (878) definition, 1V.2.22 (242) ... (892) study of, IV.12.8 (888) Absolute

convergence, in a B-space, (93) Accumulation, point of, 1.4.1 (10)

(See Continuous set function and Set function) space of, additional properties, IV.

15 (878) definition, 1V.2.22 (242) ... (892) study of, IV.12.8 (888) Absolute

convergence, in a B-space, (93) Accumulation, point of, 1.4.1 (10)

**Additive set****function**.### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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