## Linear Operators: Spectral operators |

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Page 1383

With boundary conditions A, the

from the equation sin VZ = 0. Consequently, in Case A, the

numbers of the form (not)*, n > 1; in Case C, the numbers {(n+}):t)*, n > 0.

With boundary conditions A, the

**eigenvalues**are consequently to be determinedfrom the equation sin VZ = 0. Consequently, in Case A, the

**eigenvalues**A are thenumbers of the form (not)*, n > 1; in Case C, the numbers {(n+}):t)*, n > 0.

Page 1497

In the former case the matrix B(A) necessarily has an eigenvector belonging to

the

case, to - Ao necessarily has a periodic solution, in the latter case, an anti-

periodic ...

In the former case the matrix B(A) necessarily has an eigenvector belonging to

the

**eigenvalue**+1; in the latter case, to the**eigenvalue**— 1. Thus, in the formercase, to - Ao necessarily has a periodic solution, in the latter case, an anti-

periodic ...

Page 1615

Reference: Rosenfeld, N. S., The

Operators, Comm. Pure Appl. Math. 13, 395–405 (1960). He proves the following

theorem. THEOREM. Let q(t) < 0 be twice continuously differentiable, lim, so q(t) ...

Reference: Rosenfeld, N. S., The

**Eigenvalues**of a Class of Singular DifferentialOperators, Comm. Pure Appl. Math. 13, 395–405 (1960). He proves the following

theorem. THEOREM. Let q(t) < 0 be twice continuously differentiable, lim, so q(t) ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero