## Linear Operators: Spectral operators |

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Page 1207

It is clear from Zorn's lemma that there is a maximal set A in S) for which the

spaces S3, a e A, are

that no a #0 is

to ...

It is clear from Zorn's lemma that there is a maximal set A in S) for which the

spaces S3, a e A, are

**orthogonal**. Thus to prove the lemma it suffices to observethat no a #0 is

**orthogonal**to each of the spaces S5, . Indeed, if a # 0 is**orthogonal**to ...

Page 1227

)+(Td, Tod, ) = (d, d)+(Td, id.) = (d, d.)+(d, iTod,) = (d, d.1)+(d. iTod, ) = (d, d.)+(d. i*

d, ) = 0. Similarly, (d, d_)* = 0. Next, (d_, d.)* = (d_, d, )+(T*d_, Tod, ) F- (d_, di)+(−

id_, id.) = 0. Hence the spaces Q(T), 3), and Q are mutually

)+(Td, Tod, ) = (d, d)+(Td, id.) = (d, d.)+(d, iTod,) = (d, d.1)+(d. iTod, ) = (d, d.)+(d. i*

d, ) = 0. Similarly, (d, d_)* = 0. Next, (d_, d.)* = (d_, d, )+(T*d_, Tod, ) F- (d_, di)+(−

id_, id.) = 0. Hence the spaces Q(T), 3), and Q are mutually

**orthogonal**, and 3(T) ...Page 1262

Then there exists a Hilbert space S, D \, and an

such that Ar = PQr, are \), P denoting the

{T,} be a sequence of bounded operators in Hilbert space X). Then there exists a

...

Then there exists a Hilbert space S, D \, and an

**orthogonal**projection Q in Qisuch that Ar = PQr, are \), P denoting the

**orthogonal**projection of S), on S). 29 Let{T,} be a sequence of bounded operators in Hilbert space X). Then there exists a

...

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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