## Linear Operators: Spectral theory |

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Page 906

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

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

if (Tw, w) > 0 for every a in Y); and

for ...

A bounded operator T in Hilbert space $) 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 (Tw, w) > 0 for every a in Y); 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 a(T) is a subset of the interval [0, oo). PRoof. Let E be theresolution ...

Page 1338

Let {uu) be a

respect to a

by the equations pose) = sm (A)p(dn), where e is any bounded Borel set, then the

...

Let {uu) be a

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

**positive**o-finite measure u. If the matria of densities {mu} is definedby the equations pose) = sm (A)p(dn), where e is any bounded Borel set, then the

...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

38 other sections not shown

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