Linear Operators: Spectral theory |
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Page 1099
... sufficient to consider the case in which the Hilbert space is finite - dimensional . = ( TT * ) 1/2 The argument in this special case is as follows . Since both sides of ( 1 ) are continuous in T and since every finite matrix may be ap ...
... sufficient to consider the case in which the Hilbert space is finite - dimensional . = ( TT * ) 1/2 The argument in this special case is as follows . Since both sides of ( 1 ) are continuous in T and since every finite matrix may be ap ...
Page 1475
... sufficient to prove that o ( ' , λ ) has a zero in ( a , z1 ] , for then it will follow by symmetry that σ ( :, 2 ) ... sufficiently small so that σ ( ~ ; +8 , 2。) and σ ( ≈ , —8 , 20 ) have opposite signs . Thus , ( Lemma 42 ) for ...
... sufficient to prove that o ( ' , λ ) has a zero in ( a , z1 ] , for then it will follow by symmetry that σ ( :, 2 ) ... sufficiently small so that σ ( ~ ; +8 , 2。) and σ ( ≈ , —8 , 20 ) have opposite signs . Thus , ( Lemma 42 ) for ...
Page 1684
... sufficient to prove the present lemma for the special case m = 0. By Corollary 2 again , each derivative g of order 1 of F belongs to L , ( E ) ( and has compact carrier ) , for every p ' satisfying the inequality ( i ) ( k - 1 ) 1 p ...
... sufficient to prove the present lemma for the special case m = 0. By Corollary 2 again , each derivative g of order 1 of F belongs to L , ( E ) ( and has compact carrier ) , for every p ' satisfying the inequality ( i ) ( k - 1 ) 1 p ...
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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