## Linear operators: Spectral operators |

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Page 2021

The

real and imaginary parts of a holomorphic function, has no non-zero solution <p

in &2. For, by classical function theory, any solution <p = (<px, <pa) in <Z> 2 of ...

The

**equation**.4<p = 0, being equivalent to the Cauchy-Riemann**equation**for thereal and imaginary parts of a holomorphic function, has no non-zero solution <p

in &2. For, by classical function theory, any solution <p = (<px, <pa) in <Z> 2 of ...

Page 2074

Now let y be an arbitrary vector in § + and define the vector a; by the

). Then (31) shows that x is in §+ and

some vector z in §_ we have e»'-V = e-"e-«', + >a; + «, and, using (30), it is seen ...

Now let y be an arbitrary vector in § + and define the vector a; by the

**equation**(36). Then (31) shows that x is in §+ and

**equation**(35) holds. This means that forsome vector z in §_ we have e»'-V = e-"e-«', + >a; + «, and, using (30), it is seen ...

Page 2401

Taking C7 to be of this form, we see that

nBtoiAJ -9{A1), or, using hypothesis (b), to (4) <p(B)-r(B)9(A1)=<p(Al). Using

hypothesis (c), we ...

Taking C7 to be of this form, we see that

**equation**(1) is equivalent to the**equation**(2) V + r{B))(T + <P[A1)) = T{I + r{B)), that is, to (3) r(B)T - Tr(B) = -nBtoiAJ -9{A1), or, using hypothesis (b), to (4) <p(B)-r(B)9(A1)=<p(Al). Using

hypothesis (c), we ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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