## Linear Operators: Spectral operators |

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Page 2021

Here we have ( 22 ) Â ( s ) = + i ( 8 ) s = ( 82 , 89 ) € R2 . 182 S1 For each s in R2

this matrix is normal and ( 23 ) Â ( s ) Â * ( 8 ) = 1821 = Â * ( s ) Â ( s ) , so that 8 €

R2 , A ( s ) ( A ( 24 ) 1 , 0 + 8 € R2 18 ! | 181

...

Here we have ( 22 ) Â ( s ) = + i ( 8 ) s = ( 82 , 89 ) € R2 . 182 S1 For each s in R2

this matrix is normal and ( 23 ) Â ( s ) Â * ( 8 ) = 1821 = Â * ( s ) Â ( s ) , so that 8 €

R2 , A ( s ) ( A ( 24 ) 1 , 0 + 8 € R2 18 ! | 181

**Equation**( 24 ) shows that ( sl - Â ( s )...

Page 2073

... and it follows from ( 28 ) that ( 31 ) 85 ( + ) H : CHA e - 70 + ) & H + . Now

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

+ 20 and let us write this

33 ) ...

... and it follows from ( 28 ) that ( 31 ) 85 ( + ) H : CHA e - 70 + ) & H + . Now

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

**equation**( 22 ) , that is , ( 32 ) y = a+ 20 and let us write this

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

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 z in H - we have 8868 - y = e - Be - 360 + ) ąc + 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 z in H - we have 8868 - y = e - Be - 360 + ) ąc + 2 , and , using ( 30 )

...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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adjoint operator analytic apply arbitrary assume B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded operator Chapter clear closed commuting compact complex consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator discrete domain elements equation equivalent established example exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator Math Moreover multiplicity norm positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniform uniformly unique valued vector weakly zero