## Linear Operators: Spectral operators |

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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)-1 exists as an

everywhere defined

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)-1 exists as an

everywhere defined

**bounded operator**. For 2 in p(T) the symbol R(A; T) will be ...Page 1190

An

dense so that To" is defined then the notion of symmetry is equivalent to the

inclusion to D T. Of course if T is a

the ...

An

**operator**T may be symmetric without having a dense domain but if Q(T) isdense so that To" is defined then the notion of symmetry is equivalent to the

inclusion to 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 ...

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 A suchthat the inverse

**operator**(T –ÅI)exists and is**bounded**on its domain. The set y(T)is ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### Common terms and phrases

adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero