## 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 I ( 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 I ( T ) which shows that T is

**closed**...Page 1393

Let T be a

such that the range of 11 –T is not

and is denoted by 0 , ( T ) . It is clear that 0 , ( T ) COT ) . If t is a formal differential

...

Let T be a

**closed**operator in Hilbert space . Then the set of complex numbers àsuch that the range of 11 –T is not

**closed**is called the essential spectrum of Tand is denoted by 0 , ( T ) . It is clear that 0 , ( T ) COT ) . If t is a formal differential

...

Page 1902

... 1.6.9 ( 20 ) Cauchy integral formula , ( 227 ) for functions of an operator , in a

finite dimensional space , VII.1.10 ( 560 ) in general space , VII.3.9 ( 568 )

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

Cauchy ...

... 1.6.9 ( 20 ) Cauchy integral formula , ( 227 ) for functions of an operator , in a

finite dimensional space , VII.1.10 ( 560 ) in general space , VII.3.9 ( 568 )

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

**closed**operators , VII.9.4 ( 601 )Cauchy ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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additive 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 singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero