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 ...
... 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 ...
Page 860
... algebra L1 ( —∞∞ , ∞∞ ) 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 ...
... algebra L1 ( —∞∞ , ∞∞ ) 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 ...
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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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(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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero