## Linear Operators, Part 2 |

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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 O Hwhich maps ( x , y ) into [ y , x ] then I ( T - 1 ) = A , ( T ) which shows that T is ...

Page 1226

Q . E . D . It follows from Lemma 6 ( b ) that any symmetric operator with dense

domain has a unique minimal

make the following definition . 7 DEFINITION . The minimal

Q . E . D . It follows from Lemma 6 ( b ) that any symmetric operator with dense

domain has a unique minimal

**closed**symmetric extension . This fact leads us tomake the following definition . 7 DEFINITION . The minimal

**closed**symmetric ...Page 1393

Let T be a

such that the range of 21 - T is not

and is denoted by 0 , ( T ) . It is clear that o ( T ) Co ( T ) . If r is a formal differential

...

Let T be a

**closed**operator in Hilbert space . Then the set of complex numbers àsuch that the range of 21 - T is not

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

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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 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 give 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