## Linear Operators: Spectral theory |

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Page 1120

Throughout the present section, we

space is separable. Subdiagonal representations of an operator are connected

with the study of its invariant subspaces. Thus, the key to the situation that we ...

Throughout the present section, we

**assume**for simplicity of statement that Hilbertspace is separable. Subdiagonal representations of an operator are connected

with the study of its invariant subspaces. Thus, the key to the situation that we ...

Page 1593

The functions p and q are

to be positive. The following statements describe situations in which the essential

spectrum is void: (1) For some (real or complex) A, the equation (2–1)f = 0 has ...

The functions p and q are

**assumed**to be real and continuous, and p is**assumed**to be positive. The following statements describe situations in which the essential

spectrum is void: (1) For some (real or complex) A, the equation (2–1)f = 0 has ...

Page 1594

(6) In the interval [a, b) (b s oc)

piecewise continuous in the interval [0, oo), (b) the solutions of the differential

equation d” f(t) dt? + g(t)f(t) = 0 (0 < t < 00) have only a finite number of zeros, (c)

the ...

(6) In the interval [a, b) (b s oc)

**assume**(a) the function g is non-negative andpiecewise continuous in the interval [0, oo), (b) the solutions of the differential

equation d” f(t) dt? + g(t)f(t) = 0 (0 < t < 00) have only a finite number of zeros, (c)

the ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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### Common terms and phrases

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