## Linear Operators: General theory |

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Page 479

A linear

and only if its

When these inverses exist , ( T - 1 ) * = ( T * ) - 1 . PROOF . If T - 1 exists and is in

B ...

A linear

**operator**T in B ( X , Y ) has a bounded inverse T - 1 defined on all of Y ifand only if its

**adjoint**T * has a bounded inverse ( T * ) - 1 defined on all of X * .When these inverses exist , ( T - 1 ) * = ( T * ) - 1 . PROOF . If T - 1 exists and is in

B ...

Page 612

For example , Weyl [ 2 ] showed that if T is a self

have void point spectrum ) on a separable Hilbert space , then a self adjoint

compact operator K may be found such that T + K has enough eigenvectors to

span the ...

For example , Weyl [ 2 ] showed that if T is a self

**adjoint operator**( which mayhave void point spectrum ) on a separable Hilbert space , then a self adjoint

compact operator K may be found such that T + K has enough eigenvectors to

span the ...

Page 794

Self -

Akad . Nauk SSSR 4 , 53 – 104 ( 1940 ) . ( Russian . English summary ) Math .

Rev . 2 , 104 ( 1941 ) . 8 . Spectral functions of a symmetric

Akad .

Self -

**adjoint**extensions of the second kind of a symmetric**operator**. IzvestiyaAkad . Nauk SSSR 4 , 53 – 104 ( 1940 ) . ( Russian . English summary ) Math .

Rev . 2 , 104 ( 1941 ) . 8 . Spectral functions of a symmetric

**operator**. IzvestiyaAkad .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero