## Linear Operators: Spectral theory |

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Page 929

Invariant

linear

is called a (non-trivial) invariant

Invariant

**subspaces**. If T is an operator in a J?-space £, and if 9Ji is a closedlinear

**subspace**which is neither {0} nor $ for which we have TSR Q 3Jt, then 3J?is called a (non-trivial) invariant

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

There is a one-to-one correspondence between closed symmetric

of the Hilbert space 3)(T*) which contain "£(7) and closed ... Conversely, if © is a

closed symmetric

There is a one-to-one correspondence between closed symmetric

**subspaces**©of the Hilbert space 3)(T*) which contain "£(7) and closed ... Conversely, if © is a

closed symmetric

**subspace**of 35(7'*) including 35(7), put S1 = © n (35+ ffi 35_).### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero