## Linear Operators: Spectral theory |

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Page 1187

Just as in the case of a

is defined to be the set of all complex numbers 2 such that (AI–T)- exists as an

everywhere defined

used ...

Just as in the case of a

**bounded operator**the resolvent set p(T) of an operator Tis defined to be the set of all complex numbers 2 such that (AI–T)- exists as an

everywhere defined

**bounded operator**. For 2 in p(T) the symbol R(A; T) will beused ...

Page 1190

An

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

the ...

An

**operator**T may be symmetric without having a dense domain but if 3 (T) isdense 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 1273

Weyl [5] showed this to be the case for differential

that the inverse

is called ...

Weyl [5] showed this to be the case for differential

**operators**. If T is a linear**operator**with dense domain, let y(T) be the set of all complex numbers à suchthat the inverse

**operator**(T-AI)exists and is**bounded**on its domain. The set y(T)is called ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

17 other sections not shown

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### Common terms and phrases

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