Linear Operators, Part 2 |
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Page 1000
... clear . Unfortunately it is not clear that the sequence f , is uniformly con- vergent on any region containing an interval of the real axis and so an additional argument is needed . Let U be the open interval ( a , b ) and Q the ...
... clear . Unfortunately it is not clear that the sequence f , is uniformly con- vergent on any region containing an interval of the real axis and so an additional argument is needed . Let U be the open interval ( a , b ) and Q the ...
Page 1298
... clear that if g is in D ( T1 ) , then fig and fag are also in D ( T1 ) . Since the map gfg of D ( T ) into itself is clearly closed , it is , by the closed graph theorem ( II.2.4 ) , continuous . Let B be a boundary value for 7 and ...
... clear that if g is in D ( T1 ) , then fig and fag are also in D ( T1 ) . Since the map gfg of D ( T ) into itself is clearly closed , it is , by the closed graph theorem ( II.2.4 ) , continuous . Let B be a boundary value for 7 and ...
Page 1652
... clear that { F } converges to some Fin L ( I ) . Similarly , since | FF , for each F in H ( * ) ( I ) and each index J such that J≤k , it is clear that if J≤k , the sequence { F } converges to some F , in L2 ( I ) . Let q be in Co ( I ) ...
... clear that { F } converges to some Fin L ( I ) . Similarly , since | FF , for each F in H ( * ) ( I ) and each index J such that J≤k , it is clear that if J≤k , the sequence { F } converges to some F , in L2 ( I ) . Let q be in Co ( I ) ...
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 domain 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 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 tr(T transform unique unitary vanishes vector zero