## Linear Operators: Spectral theory |

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Page 1378

matrix measure { fus } , i , j = 1 , ... , k of

, ... , k ; Pij = 0 , if i > k or ; > k . Proof . Suppose that 01 , ... , 01 is a determining set

for T. Then it is evident from

matrix measure { fus } , i , j = 1 , ... , k of

**Theorem**23 is unique , and Pii Pij , i , j = 1, ... , k ; Pij = 0 , if i > k or ; > k . Proof . Suppose that 01 , ... , 01 is a determining set

for T. Then it is evident from

**Theorem**23 that if we define { Pus } , i , j = 1 , ... , n ...Page 1379

{ ôij } is the matrix measure of

determined for each e C N. Since 1 is the union ... Then if , for i > k , the functions

0 ; ( a ) of

extended to ...

{ ôij } is the matrix measure of

**Theorem**23 , the values Pis ( e ) are uniquelydetermined for each e C N. Since 1 is the union ... Then if , for i > k , the functions

0 ; ( a ) of

**Theorem**18 ( or , the functions 05 ; ( 2 ) of**Theorem**18 ) may beextended to ...

Page 1904

( See also Ergodic

, VII.5.32 ( 584 ) by inverting sequences , VIII.2.13 ( 650 ) study of , VII.7 for

kernels , III.12.10-12 ( 219-222 ) Lebesgue dominated convergence

3.7 ...

( See also Ergodic

**theorems**) in general spaces , VII.3.13 ( 571 ) , VII.3.23 ( 576 ), VII.5.32 ( 584 ) by inverting sequences , VIII.2.13 ( 650 ) study of , VII.7 for

kernels , III.12.10-12 ( 219-222 ) Lebesgue dominated convergence

**theorem**, III.3.7 ...

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