## Linear Operators: Spectral theory |

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Page 930

It is not known whether every operator, distinct from the zero and identity

operators, has a non-trivial invariant subspace. It is readily seen from Theorem

VII.3.10 that if T is a bounded

at least ...

It is not known whether every operator, distinct from the zero and identity

operators, has a non-trivial invariant subspace. It is readily seen from Theorem

VII.3.10 that if T is a bounded

**linear operator**in a B-space 3; and if o(T) containsat least ...

Page 935

Every bounded

form UP where U is a partial isometry and P is a positive

then U may be taken to be unitary and such that U and P commute with each

other ...

Every bounded

**linear**transformation T in a Hilbert space can be written in theform UP where U is a partial isometry and P is a positive

**operator**. If T is normal,then U may be taken to be unitary and such that U and P commute with each

other ...

Page 1016

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. properties

of finite dimensional operators, which in the case of a general

Hilbert space are irrevocably lost, will be retained by Hilbert-Schmidt operators.

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. properties

of finite dimensional operators, which in the case of a general

**linear operator**inHilbert space are irrevocably lost, will be retained by Hilbert-Schmidt operators.

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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