Linear Operators: Spectral theory |
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Page 1220
... linearly independent in M ( S , E , v ) if and only if it is linearly independent in G ( S , Σ , v ) . 1 1 = The linear independence of W1 ( · , 2 ) , . . . , W2 ( , λ ) will be proved by induction on n . The case n = 1 is simply the ...
... linearly independent in M ( S , E , v ) if and only if it is linearly independent in G ( S , Σ , v ) . 1 1 = The linear independence of W1 ( · , 2 ) , . . . , W2 ( , λ ) will be proved by induction on n . The case n = 1 is simply the ...
Page 1306
... independent solutions of ( T - λ ) σ = 0 square integrable at a or b when I ( 2 ) 0. There are four possibilities as shown by the discussion above . Number of linearly independent solutions square - integrable : ( i ) ( ii ) ( iii ) ...
... independent solutions of ( T - λ ) σ = 0 square integrable at a or b when I ( 2 ) 0. There are four possibilities as shown by the discussion above . Number of linearly independent solutions square - integrable : ( i ) ( ii ) ( iii ) ...
Page 1311
... linearly independent boundary conditions B , ( f ) = 0 , i = 1 , ... , k ; i.e. , T is the restriction of T1 ( T ) ... linearly in- dependent boundary conditions which define T. Notice that k may actually be zero . 1 LEMMA . Let T have a ...
... linearly independent boundary conditions B , ( f ) = 0 , i = 1 , ... , k ; i.e. , T is the restriction of T1 ( T ) ... linearly in- dependent boundary conditions which define T. Notice that k may actually be zero . 1 LEMMA . Let T have a ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero