## Linear Operators, Part 2 |

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Page 909

The precise

elsewhere the symbol L . He will be used for the direct sum of the Hilbert spaces

Ha ( cf . IV . 4 . 18 and IV . 4 . 19 ) . The ath component of an element x in L . will ...

The precise

**meaning**of this phrase is given in the following definition . Here andelsewhere the symbol L . He will be used for the direct sum of the Hilbert spaces

Ha ( cf . IV . 4 . 18 and IV . 4 . 19 ) . The ath component of an element x in L . will ...

Page 1280

... means that f ' is continuous from the left ( or right ) at c . It should be remarked

that if t is a formal differential operator of order n on the interval I , and fe A " ( I ) ,

then the expression tf has a definite

...

... means that f ' is continuous from the left ( or right ) at c . It should be remarked

that if t is a formal differential operator of order n on the interval I , and fe A " ( I ) ,

then the expression tf has a definite

**meaning**in a very natural sense : we put ( rf )...

Page 1648

If the operator t of Definition 5 is 20 , where J is any index for E " , then Definition

5 assigns a

operator e is of order zero , i . e . , is the operation of multiplying by a function a in

Co ( I ) ...

If the operator t of Definition 5 is 20 , where J is any index for E " , then Definition

5 assigns a

**meaning**to DF for each F in D ( I ) . If , on the other hand , theoperator e is of order zero , i . e . , is the operation of multiplying by a function a in

Co ( I ) ...

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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