## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 1188

If the domain D ( T ) of the operator T is

If the domain D ( T ) of the operator T is

**dense**in H then the domain D ( T * ) consists , by definition , of all y in y for which ( Tx , y ) is continuous for x in D ( T ) . Since D ( T ) is**dense**in H there is ( IV.4.5 ) a uniquely ...Page 1245

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

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 , & € D ( A ) ) self adjoint ...Page 1271

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

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 ) = Tx2 + x2 2 2 . This shows that if ( T + il ) x = 0 , then x = 0 ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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### Other editions - View all

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

### Common terms and phrases

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 constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f 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