## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2241

These equations show that T ( f ) and T ( 1 / f ) are one - to - one operators and

that the range of T ( f ) is a

in D ( T ( f ) ) , then x = T ( 1 / 8 ) T ( f ) x , and so x is in the range R ( T ( 1 / 8 ) ) of

T ...

These equations show that T ( f ) and T ( 1 / f ) are one - to - one operators and

that the range of T ( f ) is a

**subset**of the domain of T ( 1 / f ) and vice versa . If x isin D ( T ( f ) ) , then x = T ( 1 / 8 ) T ( f ) x , and so x is in the range R ( T ( 1 / 8 ) ) of

T ...

Page 2256

Since o ( T ) is totally disconnected , each point , in o ( T ) is contained in an

arbitrarily small compact

of o ( T ) . It follows that the set 7 ( 0 ) = { 212 - 1€o } is a compact

...

Since o ( T ) is totally disconnected , each point , in o ( T ) is contained in an

arbitrarily small compact

**subset**o of o ( T ) which is open in the relative topologyof o ( T ) . It follows that the set 7 ( 0 ) = { 212 - 1€o } is a compact

**subset**of o ( R )...

Page 2257

Since by assumption | E ( 0 ; T ) | is uniformly bounded for o in K , since each

compact open

is uniformly bounded as oy runs over the family of all compact open

...

Since by assumption | E ( 0 ; T ) | is uniformly bounded for o in K , since each

compact open

**subset**o , of o ( R ) is ... 10 , it follows immediately that E ( 01 ; R )is uniformly bounded as oy runs over the family of all compact open

**subsets**of o...

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