Linear Operators: Spectral theory |
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Page 1039
... plane starting at the origin . Suppose that each of the five regions into which the plane is divided by these arcs is contained in a sector of angular opening less than л / 2 . Let N > 0 be an integer , and let T be a Hilbert - Schmidt ...
... plane starting at the origin . Suppose that each of the five regions into which the plane is divided by these arcs is contained in a sector of angular opening less than л / 2 . Let N > 0 be an integer , and let T be a Hilbert - Schmidt ...
Page 1451
... plane Rz1 . Since 2 is any number smaller than 20 , σ ( 7 ) lies entirely in the half - plane Rz 20. Q.E.D. 19 COROLLARY . Let t be a real self adjoint second order formal differential operator of the form T = - d p ( t ) dt dt ( 1/2 ) ...
... plane Rz1 . Since 2 is any number smaller than 20 , σ ( 7 ) lies entirely in the half - plane Rz 20. Q.E.D. 19 COROLLARY . Let t be a real self adjoint second order formal differential operator of the form T = - d p ( t ) dt dt ( 1/2 ) ...
Page 1532
... plane for any values of x , y ; Þ ( a + n , y + n ; z ) would also be exponentially small in the left half - plane ; and , if y - a > 0 , we know from our asymptotic formula [ 8 ] that this is impossible . Thus we must have D ( α , y ...
... plane for any values of x , y ; Þ ( a + n , y + n ; z ) would also be exponentially small in the left half - plane ; and , if y - a > 0 , we know from our asymptotic formula [ 8 ] that this is impossible . Thus we must have D ( α , y ...
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