## Linear Operators: Spectral operators |

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Page 1290

is formally self adjoint provided only that the

way, the formal differential operator (i/2)(d/dt)"{p(t)(d/dt)+(d/dt)p(t)}(d/dt)" is

formally self adjoint provided that p(t) is a real function. If we use these

observations ...

is formally self adjoint provided only that the

**coefficients**p, are real. In the sameway, the formal differential operator (i/2)(d/dt)"{p(t)(d/dt)+(d/dt)p(t)}(d/dt)" is

formally self adjoint provided that p(t) is a real function. If we use these

observations ...

Page 1320

If p is any other basis for the set of solutions of the equations to g = 0 which are

square-integrable in the neighborhood of a, then the

relation q- qK(t, s) = X 3,0)obo (s) = X x,(t)q}(s), s 3 ti j=1 j=1 will be related to the ...

If p is any other basis for the set of solutions of the equations to g = 0 which are

square-integrable in the neighborhood of a, then the

**coefficients**3, defined by therelation q- qK(t, s) = X 3,0)obo (s) = X x,(t)q}(s), s 3 ti j=1 j=1 will be related to the ...

Page 1435

o where, however, the infinite series constituting the final factor of this expression

is not convergent but is a divergent asymptotic series. The

c', co, . . . can be determined by formally substituting the asymptotic expression ...

o where, however, the infinite series constituting the final factor of this expression

is not convergent but is a divergent asymptotic series. The

**coefficients**k”, ..., k”, e,c', co, . . . can be determined by formally substituting the asymptotic expression ...

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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