## Linear Operators: Spectral theory |

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Page 929

Invariant

called a (non-trivial) invariant

space ...

Invariant

**subspaces**. If T is an operator in a B-space 3., and if st is a closed linear**subspace**which is neither {0} nor 3: for which we have TJR CŞst, then Jo iscalled a (non-trivial) invariant

**subspace**of 3 with respect to T. If 3 is a Hilbertspace ...

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 & is a closedsymmetric

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

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 other sections not shown

### 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 Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero