Linear Operators: Spectral theory |
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Page 860
... algebra L ( -∞∞ , ∞ ) with convolution as multiplication is a commutative algebra with an involution defined by f * ( s ) = f ( −s ) . It fails to be a B - algebra because it lacks a unit e . We shall show how a unit may be adjoined ...
... algebra L ( -∞∞ , ∞ ) with convolution as multiplication is a commutative algebra with an involution defined by f * ( s ) = f ( −s ) . It fails to be a B - algebra because it lacks a unit e . We shall show how a unit may be adjoined ...
Page 868
... 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 ) . It is readily seen that every ideal J in which contains properly determines an ideal in ...
... 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 ) . It is readily seen that every ideal J in which contains properly determines an ideal in ...
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 | 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