## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 984

The set of functions f in L ( R ) for which s

The set of functions f in L ( R ) for which s

**vanishes**in a neighborhood of infinity is dense in Ly ( R ) . PROOF . It follows from Lemma 3.6 that the set of all functions in L2 ( R , B , u ) which**vanish**outside of compact sets is ...Page 997

Let | be a function in Ly ( R ) L2 ( R ) whose transform of

Let | be a function in Ly ( R ) L2 ( R ) whose transform of

**vanishes**on the complement of V and let A be the linear manifold in L ( R ) of elements of the form n ) { cit ma Σε Qv Vy ( x ) = = c ; [ x , m ; ] i = 1 where m ...Page 1650

If F

If F

**vanishes**in each set I , it**vanishes**in U.la Proof . The proofs of the first four parts of this lemma are left to the reader as an exercise . To prove ( v ) , we must show from our hypothesis that F ( q ) = 0 if q is in CO ( UqIą ) ...### What people are saying - Write a review

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additive adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined eigenvalues element equal equation Exercise exists extension fact finite dimensional follows formal formal differential operator formula function function f given Hence Hilbert space Hilbert-Schmidt ideal identity independent inequality integral interval isometric isomorphism Lemma limit linear matrix measure multiplicity neighborhood norm normal operator obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solution spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero