Linear Operators, Part 2 |
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Page 1528
... equations for the coefficients . The first of these algebraic equations , which is simply the characteristic equation of the differential equation , is quadratic ; all the succeeding equations are linear . If we find the differential ...
... equations for the coefficients . The first of these algebraic equations , which is simply the characteristic equation of the differential equation , is quadratic ; all the succeeding equations are linear . If we find the differential ...
Page 1529
... equation has the characteristic equation a2 - α = 0 , so that ( 1 ) 0 , so that ¿ { 1 ) = 0 , 511 ) 1. Thus the Stokes lines for this equation are the positive and negative imaginary axes . Trying solutions of the form 2-1 ( 1 + c / z + ...
... equation has the characteristic equation a2 - α = 0 , so that ( 1 ) 0 , so that ¿ { 1 ) = 0 , 511 ) 1. Thus the Stokes lines for this equation are the positive and negative imaginary axes . Trying solutions of the form 2-1 ( 1 + c / z + ...
Page 1685
... equation ; Definitions 3.4 and 3.5 ( a ) to obtain the second equation ; an elemen- tary change of variable to obtain the third equation ; the facts that h ; is in L1 ( E " ) and that g is in Co ° ( E " ) to obtain the fourth equation ...
... equation ; Definitions 3.4 and 3.5 ( a ) to obtain the second equation ; an elemen- tary change of variable to obtain the third equation ; the facts that h ; is in L1 ( E " ) and that g is in Co ° ( E " ) to obtain the fourth equation ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero