## Linear Operators, Part 2 |

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Page 1187

A bounded operator is

A bounded operator is

**closed**if and only if its domain is**closed**. PROOF . If A , is the isometric automorphism in H H which maps ( x , y ) into [ y , x ] then I ( T - 1 ) = A I ( T ) which shows that T is**closed**if and only if T - 1 is ...Page 1393

Let T be a

Let T be a

**closed**operator in Hilbert space . Then the set of complex numbers à such that the range of 11 –T is not**closed**is called the essential spectrum of T and is denoted by 0 , ( T ) . It is clear that 0 , ( T ) COT ) .Page 1902

... ( 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

... ( 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 integral theorem ...### What people are saying - Write a review

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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