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

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Page 1188

If the domain D ( T ) of the operator T is

If the domain D ( T ) of the operator T is

**dense**in H then the domain D ( T * ) consists , by definition , of all y in ♡ for which ( Tx , y ) is continuous for x in D ( T ) . Since D ( T ) is**dense**in there is ( IV.4.5 ) a uniquely ...Page 1245

The Canonical Factorization = In this section we shall prove that each closed operator T with

The Canonical Factorization = In this section we shall prove that each closed operator T with

**dense**domain in Hilbert space has a unique factorization T PA where A is a positive ( i.e. , ( Ax , x ) 20 , x € D ( A ) ) self adjoint ...Page 1246

We may also regard A as a mapping from the

We may also regard A as a mapping from the

**dense**subspace D ( T ) of H into the space Hz . In this case A is still continuous , for | Axli = ( Ax , Ax ) , = ( A2x , x ) , ( A2 x , x ) , = ( Ax , x ) , x + 2 ( T ) , and , by the ...### 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