Linear Operators: Spectral theory |
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Page 1432
... zero for 0 ≤ a≤ n , and where we suppose that v is minimal ; i.e. , that the differential equation [ * ] does not have the form oẞx ( t ) t ̄μ ( n − k ) f ( k ) ( z ) = 0 where ẞ is analytic in the neighborhood of zero for 0 ≤ k ...
... zero for 0 ≤ a≤ n , and where we suppose that v is minimal ; i.e. , that the differential equation [ * ] does not have the form oẞx ( t ) t ̄μ ( n − k ) f ( k ) ( z ) = 0 where ẞ is analytic in the neighborhood of zero for 0 ≤ k ...
Page 1474
... zero between every pair of zeros of o ( t , 21 ) , we have only to show that the interval ( a , z ] between a and the smallest zero z of o ( t , 21 ) contains a zero of o ( t , 2 ) , and we will have established that o ( t , 2 ) has at ...
... zero between every pair of zeros of o ( t , 21 ) , we have only to show that the interval ( a , z ] between a and the smallest zero z of o ( t , 21 ) contains a zero of o ( t , 2 ) , and we will have established that o ( t , 2 ) has at ...
Page 1475
... zero in ( a , z1 ] and a zero in [ zą , b ) , we will have established that σ ( ́ , ¿ 1⁄2 ) has at least n + 1 zeros in ( a , b ) , contradicting the fact that A is in Jn . It is sufficient to prove that o ( ' , λ ) has a zero in ( a ...
... zero in ( a , z1 ] and a zero in [ zą , b ) , we will have established that σ ( ́ , ¿ 1⁄2 ) has at least n + 1 zeros in ( a , b ) , contradicting the fact that A is in Jn . It is sufficient to prove that o ( ' , λ ) has a zero in ( a ...
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