## Linear Operators: Spectral operators |

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Page 929

Invariant

a (non-trivial) invariant

Invariant

**subspaces**. If T is an operator in a B-space 3., and if Jo is a closed linear**subspace**which is neither {0} nor & for which we have Toll CŞR, then R is calleda (non-trivial) invariant

**subspace**of 3 with respect to T. If 3 is a Hilbert space ...Page 930

this is far from clear, and it is of considerable interest to find non-trivial invariant

from the zero and identity operators, has a non-trivial invariant

this is far from clear, and it is of considerable interest to find non-trivial invariant

**subspaces**for a given operator. It is not known whether every operator, distinctfrom the zero and identity operators, has a non-trivial invariant

**subspace**.Page 1228

(a) The space 3 is symmetric if and only if &l is the graph of an isometric

transformation mapping a

restriction of To to 3 is self adjoint if and only if &" is the graph of an isometric

transformation ...

(a) The space 3 is symmetric if and only if &l is the graph of an isometric

transformation mapping a

**subspace**of Q, onto a**subspace**of 3) . (b) Therestriction of To to 3 is self adjoint if and only if &" is the graph of an isometric

transformation ...

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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