## Linear Operators: Spectral theory |

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Page 893

In summary we state the following theorem . 1 THEOREM . Let E be a

self adjoint spectral measure in Hilbert space defined on a field of subsets of a

set S. Then the map Í → T ( f ) defined by the equation T ( f ) = $ ( 8 ) E ( ds ) , fe B

( S ...

In summary we state the following theorem . 1 THEOREM . Let E be a

**bounded**self adjoint spectral measure in Hilbert space defined on a field of subsets of a

set S. Then the map Í → T ( f ) defined by the equation T ( f ) = $ ( 8 ) E ( ds ) , fe B

( S ...

Page 900

and thus there is a

set having E measure zero . If f is E - measurable then to is a

measurable function , i.e. , an element of the B * -algebra B ( S , E ) . The algebra

EB ( S ...

and thus there is a

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

**bounded**E -measurable function , i.e. , an element of the B * -algebra B ( S , E ) . The algebra

EB ( S ...

Page 1240

Semi -

extensions of those operators in a class of symmetric operators which arise

frequently from the boundary value problems of mathematical physics . 1

DEFINITION .

Semi -

**bounded**Symmetric Operators In this section we study the self adjointextensions of those operators in a class of symmetric operators which arise

frequently from the boundary value problems of mathematical physics . 1

DEFINITION .

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