## Linear Operators: Spectral theory |

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Page 1187

The inverse of a

only if its domain is

which maps ( x , y ) into [ y , x ] then I ( T - 1 ) = A [ ' ( T ) which shows that T is

The inverse of a

**closed**operator is**closed**. A bounded operator is**closed**if andonly if its domain is

**closed**. Proof . If A , is the isometric automorphism in H + Hwhich maps ( x , y ) into [ y , x ] then I ( T - 1 ) = A [ ' ( T ) which shows that T is

**closed**...Page 1393

We begin by defining a certain type of “ spectrum ” for the formal differential

operator t . 1 DEFINITION . Let T be a

set of complex numbers such that the range of 11 - T is not

We begin by defining a certain type of “ spectrum ” for the formal differential

operator t . 1 DEFINITION . Let T be a

**closed**operator in Hilbert space . Then theset of complex numbers such that the range of 11 - T is not

**closed**is called the ...Page 1902

9 ( 568 ) remarks on , ( 607 - 609 ) , ( 612 ) for unbounded

9 . 4 ( 601 ) Cauchy integral theorem , ( 225 ) Cauchy problem , ( 613 - 614 ) , (

639 - 641 ) Cauchy sequence , generalized , ( 28 ) in a metric space , 1 . 6 .

9 ( 568 ) remarks on , ( 607 - 609 ) , ( 612 ) for unbounded

**closed**operators , VII .9 . 4 ( 601 ) Cauchy integral theorem , ( 225 ) Cauchy problem , ( 613 - 614 ) , (

639 - 641 ) Cauchy sequence , generalized , ( 28 ) in a metric space , 1 . 6 .

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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### Common terms and phrases

additive adjoint operator Akad 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 Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero