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

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**positive**if it is self adjoint and if ( Tx , x ) 20 for every x in H ; and**positive**definite if it is**positive**and ( Tx , x ) > 0 for every x + 0 in H. It is clear that all of these classes of operators are normal .Page 1247

Q.E.D. z Next we shall require some information on

Q.E.D. z Next we shall require some information on

**positive**self adjoint transformations and their square roots . 9 2 LEMMA . A self adjoint transformation T is**positive**if and only if o ( T ) is a ...Page 1338

Let { uis } be a

Let { uis } be a

**positive**matrix measure whose elements Mis are continuous with respect to a**positive**o - finite measure u . If the matrix of densities { m } is defined by the equations His ( e ) = S.m , ( ) u ( da ) , Mijle ) where e ...### 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