## Linear Operators: Spectral theory |

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Page 915

Hilbert space Sy and let E

sequence {a,) C. § such that § = X.1 $5(x,), where $3 (r) = SE {f(T)a,fe C(a(T))},

and a decreasing sequence {e,} of Borel sets such that (E(e)w, w,) = (E(ee,)ri, arı),

...

Hilbert space Sy and let E

**denote**its resolution of the identity. Then there erists asequence {a,) C. § such that § = X.1 $5(x,), where $3 (r) = SE {f(T)a,fe C(a(T))},

and a decreasing sequence {e,} of Borel sets such that (E(e)w, w,) = (E(ee,)ri, arı),

...

Page 1126

of the closed set C; we shall

Since each projection in the spectral resolution of T and hence each continuous

function of T is a strong limit of linear combinations of the projections E, ...

of the closed set C; we shall

**denote**this subspace of L2[0, 1] by the symbol L.(C ).Since each projection in the spectral resolution of T and hence each continuous

function of T is a strong limit of linear combinations of the projections E, ...

Page 1486

In the next few paragraphs t

operator of order n, defined on the interval R = {– o – t < -i- oc). Let t have the form

n d j -3.0 (). and suppose that all the coefficients a, are periodic and have the

same ...

In the next few paragraphs t

**denotes**a formally self adjoint formal differentialoperator of order n, defined on the interval R = {– o – t < -i- oc). Let t have the form

n d j -3.0 (). and suppose that all the coefficients a, are periodic and have the

same ...

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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