## Linear Operators: Spectral operators |

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Page 2162

Thus , in view of

that T has property ( D ) . According to Lemma 10 condition ( D ) will be satisfied if

the points regular relative to T are dense on lo . Thus Lemmas 12 , 13 , 14 give ...

Thus , in view of

**Theorem**4 . 5 , to prove the present**theorem**it suffices to showthat T has property ( D ) . According to Lemma 10 condition ( D ) will be satisfied if

the points regular relative to T are dense on lo . Thus Lemmas 12 , 13 , 14 give ...

Page 2249

... and it follows immediately from

This proves the first two assertions of the present

o ( f ( T ) ) = f ( o ( T ) ) , which is part of the final assertion of the present

... and it follows immediately from

**Theorem**9 ( v ) that ( 1 / f ) ( A ) is bounded .This proves the first two assertions of the present

**theorem**, and also the inclusiono ( f ( T ) ) = f ( o ( T ) ) , which is part of the final assertion of the present

**theorem**.Page 2418

exists almost everywhere for each fe Īp ( D , Y ) , and , using Lemma 5 once more

, the integral ( 74 ) 5 , 14 , 62 " , zv { S , 14 , | 15 ( 2 ) | dx dy Jdx dy exists for

almost all z " e D . Therefore , by Tonelli ' s

75 ) ...

exists almost everywhere for each fe Īp ( D , Y ) , and , using Lemma 5 once more

, the integral ( 74 ) 5 , 14 , 62 " , zv { S , 14 , | 15 ( 2 ) | dx dy Jdx dy exists for

almost all z " e D . Therefore , by Tonelli ' s

**theorem**( III . 11 . 14 ) , the integral (75 ) ...

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