## Linear Operators: Spectral theory |

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Page 891

Let X be a field of subsets of a

algebra of bounded linear operators on the B-space 3. It is assumed that E is

Let X be a field of subsets of a

**set**S and let E be a**function**which maps X into thealgebra of bounded linear operators on the B-space 3. It is assumed that E is

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

Hence if e1 and e, are disjoint then p(ei U ex) = E(ei U eg)"p(ei U ex) = [E(e1)+E(

eg)]"p(ei U e2) = E(e)y(ei U ex)+E(eg) p(ei U eg) = p(e1)+(p(eg), so that the

vector valued

Hence if e1 and e, are disjoint then p(ei U ex) = E(ei U eg)"p(ei U ex) = [E(e1)+E(

eg)]"p(ei U e2) = E(e)y(ei U ex)+E(eg) p(ei U eg) = p(e1)+(p(eg), so that the

vector valued

**set function**p is**additive**on 30. Therefore, if e1 on e2 = b, the**set**...Page 1899

(See Continuous

15 (378) definition, IV.2.22 (242) remarks ... additivity of regular

compact spaces, III.5.13 (138) on C(S) convergence of bounded

(See Continuous

**set function**and**Set function**) space of, additional properties, IV.15 (378) definition, IV.2.22 (242) remarks ... additivity of regular

**set functions**oncompact spaces, III.5.13 (138) on C(S) convergence of bounded

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

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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