## Linear Operators, Part 2 |

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Page 1037

1,, it is readily seen that the function <p,1(T) is

only for 2. in o(T). It remains to show that if 7. qé 0, then <p,\(T) is continuous in T

relative to the Hilbert-Schmidt norm in HS. To do this let {Tn} be a sequence in

HS ...

1,, it is readily seen that the function <p,1(T) is

**analytic**for 1 qé 0 and vanishesonly for 2. in o(T). It remains to show that if 7. qé 0, then <p,\(T) is continuous in T

relative to the Hilbert-Schmidt norm in HS. To do this let {Tn} be a sequence in

HS ...

Page 1040

y1(/1) is

; T)*y vanishes which will prove that y(}.) is

y(1) can only fail to be

y1(/1) is

**analytic**even at J. = 1,". It will now be shown that 3/2(1) = ).NE(Im; T)*R(h; T)*y vanishes which will prove that y(}.) is

**analytic**at all the points 1 = 1,", so thaty(1) can only fail to be

**analytic**at the point 1 = 0. To show this, note that (y2(Z), ...Page 1102

The determinant det(I+zTn) is an

Tn operates in finite-dimensional space, and hence more generally if Tn has a

finite-dimensional range. Thus, since a bounded convergent sequence of

The determinant det(I+zTn) is an

**analytic**(and even a polynomial) function of z, ifTn operates in finite-dimensional space, and hence more generally if Tn has a

finite-dimensional range. Thus, since a bounded convergent sequence of

**analytic**...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero