## Linear Operators: Spectral theory |

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Page 893

Let E be a

field 2 of subsets of a set S. Then the map f —- T(f) defined by the equation T(s) =

s.ss)B(is), fe B(S. 2), is a continuous *-homomorphic map of the B"-algebra B(S, ...

Let E be a

**bounded**self adjoint spectral measure in Hilbert space defined on afield 2 of subsets of a set S. Then the map f —- T(f) defined by the equation T(s) =

s.ss)B(is), fe B(S. 2), is a continuous *-homomorphic map of the B"-algebra B(S, ...

Page 900

and thus there is a

having E measure zero. If f is 2-measurable then fo is a

function, i.e., an element of the B"-algebra B(S, X). The algebra EB(S, X) of ...

and thus there is a

**bounded**function fo on S with f(s) = fo(s) except for s in a sethaving E measure zero. If f is 2-measurable then fo is a

**bounded**2-measurablefunction, i.e., an element of the B"-algebra B(S, X). The algebra EB(S, X) of ...

Page 1455

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. 25

DEFINITION. (a) If T is a closed symmetric operator in Hilbert space which is

interval (– od, ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. 25

DEFINITION. (a) If T is a closed symmetric operator in Hilbert space which is

**bounded**below and whose essential spectrum o,(T) does not intersect theinterval (– od, ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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

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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