Linear Operators, Part 2 |
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Page 866
... algebra X , the cosets x + I , xX form an algebra under the following definitions ( x + 3 ) + ( y + I ) = ( x + y ) + I x ( x + 3 ) = ( xx ) + I , ( x + 3 ) ( y + 3 ) = ( xy ) + I . This algebra is the quotient algebra of X by I and is ...
... algebra X , the cosets x + I , xX form an algebra under the following definitions ( x + 3 ) + ( y + I ) = ( x + y ) + I x ( x + 3 ) = ( xx ) + I , ( x + 3 ) ( y + 3 ) = ( xy ) + I . This algebra is the quotient algebra of X by I and is ...
Page 868
... 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 ) . It is readily seen that every ideal J in which contains 3 properly determines an ideal ...
... 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 ) . It is readily seen that every ideal J in which contains 3 properly determines an ideal ...
Page 979
... algebra A of the preceding section , we have met before . For convenience , its definition and some of its ... algebra under convolution as multiplication and the mapping f → T ( f ) is a con- tinuous isomorphism of the algebra L1 ( R ) ...
... algebra A of the preceding section , we have met before . For convenience , its definition and some of its ... algebra under convolution as multiplication and the mapping f → T ( f ) is a con- tinuous isomorphism of the algebra L1 ( R ) ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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(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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero