## Linear Operators: Spectral theory |

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Page 906

A bounded operator T in Hilbert space X is called unitary if TT* = To T = I; it is

called self adjoint, symmetric or Hermitian if T = To;

if (Tr, ar) > 0 for every a in S); and

for ...

A bounded operator T in Hilbert space X is called unitary if TT* = To T = I; it is

called self adjoint, symmetric or Hermitian if T = To;

**positive**if it is self adjoint andif (Tr, ar) > 0 for every a in S); and

**positive**definite if it is**positive**and (Tw, w) > 0for ...

Page 1247

Q.E.D. Next we shall require some information on

transformations and their square roots. 2 LEMMA. A self adjoint transformation T

is

resolution ...

Q.E.D. Next we shall require some information on

**positive**self adjointtransformations and their square roots. 2 LEMMA. A self adjoint transformation T

is

**positive**if and only if o(T) is a subset of the interval [0, 00). PRoof. Let E be theresolution ...

Page 1338

Let {u,} be a

respect to a

the equations 11,[e) = sm,(A)p(ix), where e is any bounded Borel set, then the ...

Let {u,} be a

**positive**matria measure whose elements p, are continuous withrespect to a

**positive**o-finite measure u. If the matria of densities {m,} is defined bythe equations 11,[e) = sm,(A)p(ix), where e is any bounded Borel set, then the ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero