## Linear Operators, Part 2 |

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Page 1013

For example , if { xn } is an orthonormal set in a separable

the operator determined by the equations Txn = n - txn , n = 1 , 2 , . . . . The

operator T is compact ( cf . Exercise X . 8 . 5 ) , but it is not in HS . It has been

noted in ...

For example , if { xn } is an orthonormal set in a separable

**Hilbert**space , let T bethe operator determined by the equations Txn = n - txn , n = 1 , 2 , . . . . The

operator T is compact ( cf . Exercise X . 8 . 5 ) , but it is not in HS . It has been

noted in ...

Page 1025

Q . E . D . Having established these preliminary theorems on finite dimensional

spaces , we now return to the study of

the notion of trace to certain operators in

Q . E . D . Having established these preliminary theorems on finite dimensional

spaces , we now return to the study of

**Hilbert**space . It is desired to generalizethe notion of trace to certain operators in

**Hilbert**space and at first glance it may ...Page 1262

Then there exists a

such that Ax = PQx , XEH , P denoting the orthogonal projection of Hi on H . 29

Let { Tn } be a sequence of bounded operators in

exists ...

Then there exists a

**Hilbert**space H , 2H , and an orthogonal projection Q in Hisuch that Ax = PQx , XEH , P denoting the orthogonal projection of Hi on H . 29

Let { Tn } be a sequence of bounded operators in

**Hilbert**space H . Then thereexists ...

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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