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 ) . 2 ) The linear independence of W1 ( · , λ ) , . . . , Wn ( ' , 2 ) 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 ) . 2 ) The linear independence of W1 ( · , λ ) , . . . , Wn ( ' , 2 ) will be proved by induction on n . The case n = 1 is simply the ...
Page 1306
... independent solutions of ( 7-2 ) 0 σ = 0 square integrable at a or b when ( 2 ) 0. There are four possibilities as shown by the discussion above . Number of linearly independent solutions square - integrable : ( i ) ( ii ) ( iii ) ( iv ) ...
... independent solutions of ( 7-2 ) 0 σ = 0 square integrable at a or b when ( 2 ) 0. There are four possibilities as shown by the discussion above . Number of linearly independent solutions square - integrable : ( i ) ( ii ) ( iii ) ( iv ) ...
Page 1311
... linearly independent boundary conditions B , ( f ) = 0 , i = 1 , ... , k ; i.e. , T is the restriction of T1 ( 7 ) ... 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 ( 7 ) ... linearly in- dependent boundary conditions which define T. Notice that k may actually be zero . 1 LEMMA . Let T have a ...
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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