Linear Operators: Spectral theory |
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Page 906
... Clearly , A and B must be given by the formulae A T + T * 2 В - 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 В - 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 fn 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 fn 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 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 lim - fm - gp = 0 and lim , dfm - gp ‚ | λ1ƒm — g3│p = 0. It is then clear from Definition 3.26 ...
... 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 lim - fm - gp = 0 and lim , dfm - gp ‚ | λ1ƒm — g3│p = 0. It is then clear from Definition 3.26 ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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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 Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero