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 ) * = αx * , ( xy ) * = y * x * ( x * ) * = x . All of the examples mentioned above , with the exception of L1 ( ∞ , ∞ ) and ... B - ALGEBRAS.
... algebra X is a mapping xx * of X into itself with the properties ( x + y ) * = x * + y * , ( xx ) * = αx * , ( xy ) * = y * x * ( x * ) * = x . All of the examples mentioned above , with the exception of L1 ( ∞ , ∞ ) and ... B - ALGEBRAS.
Page 868
Nelson Dunford, Jacob T. Schwartz. 2. Commutative B - Algebras In case is a commutative B - algebra every ideal is two - sided and the quotient algebra X / is again a commutative algebra . It will be a B - algebra if I is closed ( 1.13 ) ...
Nelson Dunford, Jacob T. Schwartz. 2. Commutative B - Algebras In case is a commutative B - algebra every ideal is two - sided and the quotient algebra X / is again a commutative algebra . It will be a B - algebra if I is closed ( 1.13 ) ...
Page 882
... algebra . 14 If f is in L ( -∞ , ∞ ) , and if λ ( E ) = √ f ( s ) ds show that ( 2 * μ ) ( E ) = √ ̧ds √∞∞ † ( s — t ) μ ( dt ) , for every μ in the space M of Exercise 13. If μ ( E ) = Se g ( s ) ds for some g in L1 ... B - ALGEBRAS.
... algebra . 14 If f is in L ( -∞ , ∞ ) , and if λ ( E ) = √ f ( s ) ds show that ( 2 * μ ) ( E ) = √ ̧ds √∞∞ † ( s — t ) μ ( dt ) , for every μ in the space M of Exercise 13. If μ ( E ) = Se g ( s ) ds for some g in L1 ... B - ALGEBRAS.
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero