Linear Operators: Spectral theory |
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Page 1217
... respectively , with measures μ and μ , and multiplicity sets { e } and { e } will be called equivalent if u ≈ μ and μ ( en ) = 0 = μ ( e̟ „ Aễn ) for n = 1 , 2 , .... 16 THEOREM . A separable Hilbert space & has an ordered ...
... respectively , with measures μ and μ , and multiplicity sets { e } and { e } will be called equivalent if u ≈ μ and μ ( en ) = 0 = μ ( e̟ „ Aễn ) for n = 1 , 2 , .... 16 THEOREM . A separable Hilbert space & has an ordered ...
Page 1302
... respectively . Then d + = d'1 + d " ―n ; d_ = d ' _ + d'_ ' —n . PROOF . Let D , be the space of solutions of tf L ( I ) , and let D + + + = if which lie in and D be the spaces of solutions of t'f = if and ' ' f if which lie in L2 ( I ...
... respectively . Then d + = d'1 + d " ―n ; d_ = d ' _ + d'_ ' —n . PROOF . Let D , be the space of solutions of tf L ( I ) , and let D + + + = if which lie in and D be the spaces of solutions of t'f = if and ' ' f if which lie in L2 ( I ...
Page 1326
... respectively , and which satisfy the boundary conditions at a and at b respectively . Then the resolvent R ( λ ; T ) = ( λI -T ) -1 is given by the expression ( R ( 2 ; T ) f ) ( t ) = √ , f ( s ) K ( t , s ; λ ) ds , where the kernel ...
... respectively , and which satisfy the boundary conditions at a and at b respectively . Then the resolvent R ( λ ; T ) = ( λI -T ) -1 is given by the expression ( R ( 2 ; T ) f ) ( t ) = √ , f ( s ) K ( t , s ; λ ) ds , where the kernel ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero