## 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 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers 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 f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero