Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 68
Page 1217
... respectively , with measures μ and ũ , and multiplicity sets { e } and { n } 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 representation ...
... respectively , with measures μ and ũ , and multiplicity sets { e } and { n } 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 representation ...
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 ...
Page 1548
... respectively by the boundary conditions f ( c ) = ƒ ' ( c ) f ( n - 1 ) ( c ) = 0 and by the boundary conditions in the set B at the right and at the left endpoints of I respectively . Show that the operators T1 and T2 are self adjoint ...
... respectively by the boundary conditions f ( c ) = ƒ ' ( c ) f ( n - 1 ) ( c ) = 0 and by the boundary conditions in the set B at the right and at the left endpoints of I respectively . Show that the operators T1 and T2 are self adjoint ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Copyright | |
52 other sections not shown
Other editions - View all
Common terms and phrases
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