## Linear Operators: Spectral operators |

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Page 905

If the spectrum of the bounded normal operator T in § is countable then there is

an

(a, y)), a E \), we B and, for each ar, all but a countable number of the coefficients

...

If the spectrum of the bounded normal operator T in § is countable then there is

an

**orthonormal basis**B for Sy consisting of eigenvectors of T. Furthermore, a = X(a, y)), a E \), we B and, for each ar, all but a countable number of the coefficients

...

Page 1028

Let {a, , o, e A} be an

may suppose without loss of generality that there is a finite subset B of A such

that {a, , o, e B} is an

Let {a, , o, e A} be an

**orthonormal basis**for $). Since ES) is finite dimensional wemay suppose without loss of generality that there is a finite subset B of A such

that {a, , o, e B} is an

**orthonormal basis**for ES), and {r., o, e A — B} is an ...Page 1029

Then, since S is necessarily invariant under T, there exists by the inductive

hypothesis, an

be orthogonal to S and have norm one so that {r}, ..., a,} is an

for E” ...

Then, since S is necessarily invariant under T, there exists by the inductive

hypothesis, an

**orthonormal basis**{r, ..., r, 1} for S with ((T-21)r, r) = 0 for j > i. Let a,be orthogonal to S and have norm one so that {r}, ..., a,} is an

**orthonormal basis**for E” ...

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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