## Linear Operators: Spectral operators |

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Page 2021

( S ) 0 88 R2 and ( 26 ) Â ( ) ¥ ( ) = 18 / 14 ( s ) ] , SES , WE H2 . The

0 , being equivalent to the Cauchy - Riemann

**Equation**( 24 ) shows that | s | - - Â ( s ) is unitary for 8 + 0 . Thus ( 25 ) Å - 4 ( 6 ) =( S ) 0 88 R2 and ( 26 ) Â ( ) ¥ ( ) = 18 / 14 ( s ) ] , SES , WE H2 . The

**equation**Ap =0 , being equivalent to the Cauchy - Riemann

**equation**for the real and ...Page 2073

... and it follows from ( 28 ) that ( 31 ) e5 ( 9 + ) H + SH + , - 5 ( 9 + ) S H + . Now

suppose that the vectors x , y in H + satisfy the

+ 3C and let us write this

...

... and it follows from ( 28 ) that ( 31 ) e5 ( 9 + ) H + SH + , - 5 ( 9 + ) S H + . Now

suppose that the vectors x , y in H + satisfy the

**equation**( 22 ) , that is , ( 32 ) y = a+ 3C and let us write this

**equation**as ( 33 ) y = ax — P _ ax .**Equations**( 30 ) and...

Page 2074

Now let y be an arbitrary vector in H . and define the vector x by the

) . Then ( 31 ) shows that x is in H . and

some vector 2 in H , we have 8569 - ) y = e - Be - 569 + 3x + 2 , and , using ( 30 )

...

Now let y be an arbitrary vector in H . and define the vector x by the

**equation**( 36) . Then ( 31 ) shows that x is in H . and

**equation**( 35 ) holds . This means that forsome vector 2 in H , we have 8569 - ) y = e - Be - 569 + 3x + 2 , and , using ( 30 )

...

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