Linear Operators: Spectral theory |
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Page 906
... Clearly , A and B must be given by the formulae A = T + T * 2 B - T - T * 2i It is clear that T is normal if and only if its " real " and " imaginary " parts A and B commute . The notion of a positive operator allows us to introduce a ...
... Clearly , A and B must be given by the formulae A = T + T * 2 B - T - T * 2i It is clear that T is normal if and only if its " real " and " imaginary " parts A and B commute . The notion of a positive operator allows us to introduce a ...
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 rectangle ...
... 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 rectangle ...
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 limm - fm - gp = 0. It is then clear from Definition 3.26 that limfmg and ...
... 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 limm - fm - gp = 0. It is then clear from Definition 3.26 that limfmg and ...
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