## Linear Operators, Part 2 |

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Page 890

If the Hilbert space is infinite dimensional there is still an operational calculus for

a normal operator T with resolution of the identity E which is given by the

(vi), but in this situation it is necessary to define the integral appearing in (vi) and

...

If the Hilbert space is infinite dimensional there is still an operational calculus for

a normal operator T with resolution of the identity E which is given by the

**formula**(vi), but in this situation it is necessary to define the integral appearing in (vi) and

...

Page 1112

a1|—1,l an—1,2' - - an-1,?! anl a'n2 ' ' ' arm bnl bn2 ' ' ' bun Therefore, by

Lagrange's expansion

d a a (T det(A+zB)l¢-0 = 2 Zbifin Z 1'-1 1-1 = det ) tr(A'1B), K where y,' denotes the

...

a1|—1,l an—1,2' - - an-1,?! anl a'n2 ' ' ' arm bnl bn2 ' ' ' bun Therefore, by

Lagrange's expansion

**formula**and Cramer's**formula**for matrix inverses, we haved a a (T det(A+zB)l¢-0 = 2 Zbifin Z 1'-1 1-1 = det ) tr(A'1B), K where y,' denotes the

...

Page 1363

basis for this

in the resolution of the identity for T corresponding to (1.1, /1,) may be calculated

from the resolvent by the

basis for this

**formula**is found in Theorem XII.2.l0 which asserts that the projectionin the resolution of the identity for T corresponding to (1.1, /1,) may be calculated

from the resolvent by the

**formula**A,-s E(().1, }.,))f =- lim lim [R(2.—ie; T)—R(}.### 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