## Linear Operators: Spectral theory |

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Page 1164

Calderón and Zygmund [ 1 ] show that if the function 2

weak , continuity hypothesis , then the singular integral [ 2 ( x , y ) , 9 ( x ) = 1 .

navn ( y ) dy ( i ) exists for almost all x if f is in L ( E " ) or L ( E " ) , 00 > p > 1 , ( cf .

Calderón and Zygmund [ 1 ] show that if the function 2

**satisfies**a suitable , ratherweak , continuity hypothesis , then the singular integral [ 2 ( x , y ) , 9 ( x ) = 1 .

navn ( y ) dy ( i ) exists for almost all x if f is in L ( E " ) or L ( E " ) , 00 > p > 1 , ( cf .

Page 1385

The function e - V - 12

if 1 - kv - = 0 ; i . e . , if and only if k is positive and a = - 1 / ko . Thus , only in case (

iv ) does Tk have a non - void point spectrum , which consists of the single ...

The function e - V - 12

**satisfies**the boundary condition f ( 0 ) + kf ' ( 0 ) if and onlyif 1 - kv - = 0 ; i . e . , if and only if k is positive and a = - 1 / ko . Thus , only in case (

iv ) does Tk have a non - void point spectrum , which consists of the single ...

Page 1705

... to

is in A ( m ) ( I ) ; here Lemma 3 . 6 ( iv ) permits the use of the Leibniz formula to

show that alt89 ) = qo08€ + g ( J , ε ) for JI Sp , where , by Lemmas 3 . 22 and 3 .

... to

**satisfies**the partial differential equation aj ( ex ) ( VEP ) = če , JI = P where ĝsis in A ( m ) ( I ) ; here Lemma 3 . 6 ( iv ) permits the use of the Leibniz formula to

show that alt89 ) = qo08€ + g ( J , ε ) for JI Sp , where , by Lemmas 3 . 22 and 3 .

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present 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 unit vanishes vector zero