## Linear Operators: Spectral operators |

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Page 1260

17 (Schmidt) The non-

eigenvalues of TT", even as to multiplicity (the positive square roots of these

eigenvalues are sometimes called the characteristic numbers of T). 18 Let T be a

...

17 (Schmidt) The non-

**zero**eigenvalues of ToT are the same as the non-**zero**eigenvalues of TT", even as to multiplicity (the positive square roots of these

eigenvalues are sometimes called the characteristic numbers of T). 18 Let T be a

...

Page 1464

Then (a) is lim sup, ... t”q(t) < – (1/4), every solution of ts = 0 has an infinite

number of

identically

PRoof. According to ...

Then (a) is lim sup, ... t”q(t) < – (1/4), every solution of ts = 0 has an infinite

number of

**zeros**on [a, oo); (b) if lim inf, .s. f*q(t) > –(1/4), no solution, notidentically

**zero**, of Ts = 0 has more than a finite number of**zeros**on sa, oo).PRoof. According to ...

Page 1727

By (1) and by the definitions (2), (3), and (5) of St, it follows that (7) (Stop)(r) = 0,

p e C(E"), —k < min (L) < max (L) < k, if one of ar1, ..., a, is

let Io denote the cube Io = {a e E"|r, < 1, i = 1, ..., n}. Then for —k < min (L) < max ...

By (1) and by the definitions (2), (3), and (5) of St, it follows that (7) (Stop)(r) = 0,

p e C(E"), —k < min (L) < max (L) < k, if one of ar1, ..., a, is

**zero**. Suppose that welet Io denote the cube Io = {a e E"|r, < 1, i = 1, ..., n}. Then for —k < min (L) < max ...

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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