Linear Operators: Spectral theory |
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Page 860
... algebra X is a mapping xx * of X into itself with the properties ( x + y ) * = x * + y * , ( xx ) * = ãx * , ( xy ) * = y * x * ( x * ) * = x . - All of the examples mentioned above , with the exception of ... B 860 IX.1.1 IX . B - ALGEBRAS.
... algebra X is a mapping xx * of X into itself with the properties ( x + y ) * = x * + y * , ( xx ) * = ãx * , ( xy ) * = y * x * ( x * ) * = x . - All of the examples mentioned above , with the exception of ... B 860 IX.1.1 IX . B - ALGEBRAS.
Page 874
... algebra is a B - algebra with an involution * which satisfies the identity x * x = x2 . Besides the preceding examples , which are all commutative B * - algebras , there is the algebra B ( § ) of all ... B - ALGEBRAS Commutative B*-Algebras.
... algebra is a B - algebra with an involution * which satisfies the identity x * x = x2 . Besides the preceding examples , which are all commutative B * - algebras , there is the algebra B ( § ) of all ... B - ALGEBRAS Commutative B*-Algebras.
Page 875
... algebra B ( S ) of all bounded linear operators in Hilbert space in which the operation of involution is defined by equation ( i ) is a B * -algebra . Our chief objective in this section is to characterize commutative B * -algebras . It ...
... algebra B ( S ) of all bounded linear operators in Hilbert space in which the operation of involution is defined by equation ( i ) is a B * -algebra . Our chief objective in this section is to characterize commutative B * -algebras . It ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero