## Linear Operators: Spectral theory |

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Page 1188

We shall need to use the notion of the Hilbert space adjoint of an operator which

is not necessarily bounded and this concept is formulated in the following

definition . 4 DEFINITION . If the domain D ( T ) of the operator T is

then the ...

We shall need to use the notion of the Hilbert space adjoint of an operator which

is not necessarily bounded and this concept is formulated in the following

definition . 4 DEFINITION . If the domain D ( T ) of the operator T is

**dense**in Hthen the ...

Page 1245

The Canonical Factorization In this section we shall prove that each closed

operator T with

where A is a positive ( i . e . , ( Ax , x ) 2 0 , X E D ( A ) ) self adjoint transformation

...

The Canonical Factorization In this section we shall prove that each closed

operator T with

**dense**domain in Hilbert space has a unique factorization T = PA ,where A is a positive ( i . e . , ( Ax , x ) 2 0 , X E D ( A ) ) self adjoint transformation

...

Page 1271

Let T be a symmetric operator with domain D ( T )

) , we have | ( T + il ) x | 2 = ( Tx , Tx ) Fi ( x , Tx ) + i ( Tx , x ) + ( x , x ) = \ Tx12 + \ x |

2 2 \ x12 . This shows that if ( T + il ) x = 0 , then x = 0 and so the operators T il ...

Let T be a symmetric operator with domain D ( T )

**dense**in H . Then if x is in D ( T) , we have | ( T + il ) x | 2 = ( Tx , Tx ) Fi ( x , Tx ) + i ( Tx , x ) + ( x , x ) = \ Tx12 + \ x |

2 2 \ x12 . This shows that if ( T + il ) x = 0 , then x = 0 and so the operators T il ...

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently consider constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero