## Linear Operators: Spectral theory |

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Page 1100

matrix with the same eigenvalues as A, the statement (3) follows from the

inequality tr(VD, V"D, V") = |D., D.,. ... Since the linear functional tris continuous on

C1, and since, as we have observed in the paragraph

→ AB ...

matrix with the same eigenvalues as A, the statement (3) follows from the

inequality tr(VD, V"D, V") = |D., D.,. ... Since the linear functional tris continuous on

C1, and since, as we have observed in the paragraph

**following Lemma**9, [A, B)→ AB ...

Page 1102

... in a Hilbert space of dimension d > 00. The formula e" 4 = det(A) is valid for

finite-dimensional matrices, det(A) denoting the determinant of A. Since the

determinant of A is the product of its eigenvalues, it

we have - ...

... in a Hilbert space of dimension d > 00. The formula e" 4 = det(A) is valid for

finite-dimensional matrices, det(A) denoting the determinant of A. Since the

determinant of A is the product of its eigenvalues, it

**follows**by**Lemma**6(a) thatwe have - ...

Page 1696

By Lemma 14 there is a sequence {F,} of elements of D(I), each of which has a

carrier which is a compact subset C, of I, ... it

3.12 that there is a unique extension G of F to a distribution in D,(D) such that the

...

By Lemma 14 there is a sequence {F,} of elements of D(I), each of which has a

carrier which is a compact subset C, of I, ... it

**follows from Lemmas**13, 3.43 and3.12 that there is a unique extension G of F to a distribution in D,(D) such that the

...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete 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 identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero