## Linear Operators: Spectral theory |

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Page 970

If xe denotes the characteristic function of the set e in S, and if/ is in L2(R), then %

ef >s m ^i(-B)nI^,(7J) and / is the

sequence {xef}- Hence, by Theorem 9, xf is the

...

If xe denotes the characteristic function of the set e in S, and if/ is in L2(R), then %

ef >s m ^i(-B)nI^,(7J) and / is the

**limit**in the norm of L2(R) of the generalizedsequence {xef}- Hence, by Theorem 9, xf is the

**limit**in the norm of L2(uf 0) of the...

Page 976

It is easily seen that this theorem asserts that if / is in L2(Rn), the

, ...,<„) = (27*)-"/* lim ... f{Xl xn) N^-coJ-N J-tr • exp {—Ht1x1+ ... + tnxn)}dx1 . . . dxn

exists in the norm of L2(R"), and defines a unitary mapping in this space whose ...

It is easily seen that this theorem asserts that if / is in L2(Rn), the

**limit**p+N p+N gft, ...,<„) = (27*)-"/* lim ... f{Xl xn) N^-coJ-N J-tr • exp {—Ht1x1+ ... + tnxn)}dx1 . . . dxn

exists in the norm of L2(R"), and defines a unitary mapping in this space whose ...

Page 1124

Hence |£^„|z = \Exxn^- for each n, so that Ex„ = Evvn and E = Ex. That is, cp(E) = <

p(Ex) implies E = Ev Similarly, <p(E) ^ <p(Ex) implies E g Ex. If £„, £ are in jF and

<p(En) increases to the

Hence |£^„|z = \Exxn^- for each n, so that Ex„ = Evvn and E = Ex. That is, cp(E) = <

p(Ex) implies E = Ev Similarly, <p(E) ^ <p(Ex) implies E g Ex. If £„, £ are in jF and

<p(En) increases to the

**limit**<p{E), then it follows from what we have already ...### 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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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