## Linear Operators: Spectral theory |

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Page 929

If T is an operator in a B-space 3., and if J. is a closed linear

neither {0} nor 3: for which we have TJ C J ... invariant

respect to T, then Jē is said to reduce T. It is not difficult to see that a non-trivial

If T is an operator in a B-space 3., and if J. is a closed linear

**subspace**which isneither {0} nor 3: for which we have TJ C J ... invariant

**subspaces**of 3 withrespect to T, then Jē is said to reduce T. It is not difficult to see that a non-trivial

**subspace**...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

Q.E.D. 11 LEMMA. There is a one-to-one correspondence between closed

symmetric

symmetric

Q.E.D. 11 LEMMA. There is a one-to-one correspondence between closed

symmetric

**subspaces**& of the Hilbert space ... Conversely, if Ø is a closedsymmetric

**subspace**of Q(T*) including £(T), put & = & n (3), ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero