## Linear Operators, Part 2 |

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Page 1333

Since gn —>0 and TR(i.; T) is a bounded operator, it follows that R(1; T)g,, and

TR(}.; T)g,,

p1.

...

Since gn —>0 and TR(i.; T) is a bounded operator, it follows that R(1; T)g,, and

TR(}.; T)g,,

**converge**to zero in L,(I). Thus by Lemma 2.16 the series 22:0 (f,<p'.)<p1.

**converges**to / in the topology of C“-1(J) for each compact interval J of I . Since...

Page 1420

Suppose that {fa}

assumption (b), {fn}

let {fn}

"I -> 0If ...

Suppose that {fa}

**converges**to zero in the topology of $(T1(r)). Then, byassumption (b), {fn}

**converges**to zero in the topology of'D(T1(r+'r')). Conversely,let {fn}

**converge**to zero in the topology of 'D(T1(1'+'t')), that is, let [*1 I/..|+|Ti(T+T')/"I -> 0If ...

Page 1436

Thus {hi}

gfl} has a convergent subsequence, which proves the first part of our theorem.

Conversely, suppose that if {fn} is bounded, and {Tfn}

Thus {hi}

**converges**, so that {hi} = {hi+2f_1w:'(h,-)<p,}**converges**. Consequently, {gfl} has a convergent subsequence, which proves the first part of our theorem.

Conversely, suppose that if {fn} is bounded, and {Tfn}

**converges**, then {fa} has a ...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

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 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 hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero