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 861
... B - algebra 7 ( X ) . Since | x | = | xe | = | Txe | ≤ | e || Tx | it follows that the inverse map 1 is continuous . To see that τ is also continuous it will first be shown that T ( X ) is closed in B ( X ) . To do this the following ...
... B - algebra 7 ( X ) . Since | x | = | xe | = | Txe | ≤ | e || Tx | it follows that the inverse map 1 is continuous . To see that τ is also continuous it will first be shown that T ( X ) is closed in B ( X ) . To do this the following ...
Page 868
Nelson Dunford, Jacob T. Schwartz. 2. Commutative B - Algebras In case X is a commutative B - algebra every ideal is two - sided and the quotient algebra X / 3 is again a commutative algebra . It will be a B - algebra if is closed ( 1.13 ) ...
Nelson Dunford, Jacob T. Schwartz. 2. Commutative B - Algebras In case X is a commutative B - algebra every ideal is two - sided and the quotient algebra X / 3 is again a commutative algebra . It will be a B - algebra if is closed ( 1.13 ) ...
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