## Linear Operators: Spectral theory |

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Page 889

This means that for every sequence {6,} of disjoint

E ( U 6,)r, a e S). i-1 i-1 A spectral ... on the

iv) for every

This means that for every sequence {6,} of disjoint

**Borel sets**co co (v) X E(6,)a =E ( U 6,)r, a e S). i-1 i-1 A spectral ... on the

**Borel sets**in the plane and satisfying (iv) for every

**Borel set**6 and (v) for every sequence {6,} of disjoint**Borel sets**is ...Page 894

6.2) that a*E(6) r = 0 for every

e 3". It follows (II.3.15) that E(0) = 0. Thus if E and A are bounded additive regular

operator valued set functions defined on the

6.2) that a*E(6) r = 0 for every

**Borel set**Ö in S and every pair w, wo with a e 3, r*e 3". It follows (II.3.15) that E(0) = 0. Thus if E and A are bounded additive regular

operator valued set functions defined on the

**Borel sets**of a normal topological ...Page 913

Let E be the spectral resolution for T and let v, (e) = (E(e)y, , y,) for each

e. ... let {e,} be a sequence of

is a Borel subset of the complement e, of e, and X. v.,(e) = 0, then v.(e) = 0.

Let E be the spectral resolution for T and let v, (e) = (E(e)y, , y,) for each

**Borel set**e. ... let {e,} be a sequence of

**Borel sets**such that X-d v,(e,) = 0, and such that if eis a Borel subset of the complement e, of e, and X. v.,(e) = 0, then v.(e) = 0.

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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