## Linear Operators: Spectral theory |

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Page 1190

An operator T may be symmetric without having a dense domain but if 3 (T) is

dense so that T+ is defined then the notion of symmetry is equivalent to the

inclusion T” D T. Of course if T is a bounded

the ...

An operator T may be symmetric without having a dense domain but if 3 (T) is

dense so that T+ is defined then the notion of symmetry is equivalent to the

inclusion T” D T. Of course if T is a bounded

**everywhere**defined operator thenthe ...

Page 1233

Then (T1-A,I)- = R(20) is an

more than |A (20)|-1. Consequently, the series [+] X (2–70)"R(Ao)"+" n=0

converges if |2–Åo < ||Y(Mo). Since T, is closed, we have (T1-AI) X (A–Ao)" R(Ao)

"+"y n=0 ...

Then (T1-A,I)- = R(20) is an

**everywhere**defined, bounded operator of norm notmore than |A (20)|-1. Consequently, the series [+] X (2–70)"R(Ao)"+" n=0

converges if |2–Åo < ||Y(Mo). Since T, is closed, we have (T1-AI) X (A–Ao)" R(Ao)

"+"y n=0 ...

Page 1899

(See Field of sets) Almost

additive scalar set functions, III.1.11 (100) definition for vector-valued set

functions, IV.10.6 (322) set function, criteria for, III.4.4–5 (127–128). (See also

Variation) ...

(See Field of sets) Almost

**everywhere**(or u-almost**everywhere**) definition foradditive scalar set functions, III.1.11 (100) definition for vector-valued set

functions, IV.10.6 (322) set function, criteria for, III.4.4–5 (127–128). (See also

Variation) ...

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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