Linear Operators, Part 2 |
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Page 1000
... clear . Unfortunately it is not clear that the sequence ƒ „ is uniformly con- vergent on any region containing an interval of the real axis and so an additional argument is needed . n Let U be the open interval ( a , b ) and Q the ...
... clear . Unfortunately it is not clear that the sequence ƒ „ is uniformly con- vergent on any region containing an interval of the real axis and so an additional argument is needed . n Let U be the open interval ( a , b ) and Q the ...
Page 1689
... clear from ( i ) that { m } is a Cauchy sequence in L ( I ) for J≤k , so that there exist functions g , g in L ( I ) such that limm - fm - gp = 0 and limfm - g = 0 . It is then clear from Definition 3.26 that limfmg and limfmg for | J ...
... clear from ( i ) that { m } is a Cauchy sequence in L ( I ) for J≤k , so that there exist functions g , g in L ( I ) such that limm - fm - gp = 0 and limfm - g = 0 . It is then clear from Definition 3.26 that limfmg and limfmg for | J ...
Page 1694
... clear that Ifm0 as moo . By the preceding corollary , we may suppose without loss of generality that { f } converges in the topology of HP - 1 ( I ) to an element g . Then we clearly have Ifm - g ( o ) → 0 as m → ∞ , and since fm ( 0 ) ...
... clear that Ifm0 as moo . By the preceding corollary , we may suppose without loss of generality that { f } converges in the topology of HP - 1 ( I ) to an element g . Then we clearly have Ifm - g ( o ) → 0 as m → ∞ , and since fm ( 0 ) ...
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