Linear Operators: Spectral theory |
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Page 898
... Corollary 4 follows immediately from Theorem 1 and Corollary IX.3.15 . Q.E.D. 5 DEFINITION . The uniquely defined spectral measure associat- ed , in Corollary 4 , with the normal operator T is called the resolution of the identity for T ...
... Corollary 4 follows immediately from Theorem 1 and Corollary IX.3.15 . Q.E.D. 5 DEFINITION . The uniquely defined spectral measure associat- ed , in Corollary 4 , with the normal operator T is called the resolution of the identity for T ...
Page 1301
... corollary were false , it would follow that τ has a boundary value at a which is independent of the set Ao , ... , An - 1 and hence has at least n + 1 independent boundary values at a . But this is impossible by Corollary 22. Q.E.D. 24 ...
... corollary were false , it would follow that τ has a boundary value at a which is independent of the set Ao , ... , An - 1 and hence has at least n + 1 independent boundary values at a . But this is impossible by Corollary 22. Q.E.D. 24 ...
Page 1459
... COROLLARY . A formally positive formally symmetric formal differential operator t is finite below zero . PROOF . It is obvious from Definition 20 that t is bounded below . Thus the present corollary follows from Corollary 7 and ...
... COROLLARY . A formally positive formally symmetric formal differential operator t is finite below zero . PROOF . It is obvious from Definition 20 that t is bounded below . Thus the present corollary follows from Corollary 7 and ...
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