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

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Page 860

A B -

A B -

**algebra**X is commutative in case xy yx for all x and y in X. An involution in a B -**algebra**X is a mapping x → ** of X into itself with the properties ( x + y ) * = x * + y * , ( xy ) * = y * r * ( ax ) * ār * , ( x * ) * = X ...Page 860

A B -

A B -

**algebra**X is commutative in case xy = yx for all x and y in X. An involution in a B -**algebra**X is a mapping x + 2 * of X into itself with the properties ( x + y ) * = x * + y * , ( xy ) * = y * r * ( ax ) * = ax * , ( x * ) * = r ...Page 979

One of these algebras , namely the

One of these algebras , namely the

**algebra**A of the preceding section , we have met before . For convenience , its definition and some of its properties will be restated here . For every f in L ( R ) the convolution ( * * g ) ( x ) ...### 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 Ly(R 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 unique unit unitary vanishes vector zero