Linear Operators, Part 2 |
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Page 898
... identity for T. In order to relate this notion of the resolution of the identity with that given in Section 1 we state the following corollary . 6 COROLLARY . If E is the resolution of the identity for the normal operator T and if 8 is ...
... identity for T. In order to relate this notion of the resolution of the identity with that given in Section 1 we state the following corollary . 6 COROLLARY . If E is the resolution of the identity for the normal operator T and if 8 is ...
Page 920
... identity for T and I respectively . From Corollary 2.7 it is seen that È VEV - 1 and hence that = F ( T ) = VF ( T ) V - 1 for every bounded Borel function F. The mapping W = ̃V of H onto ΣL ( ễ ,, ) is clearly an isometry and ...
... identity for T and I respectively . From Corollary 2.7 it is seen that È VEV - 1 and hence that = F ( T ) = VF ( T ) V - 1 for every bounded Borel function F. The mapping W = ̃V of H onto ΣL ( ễ ,, ) is clearly an isometry and ...
Page 1717
... identity 2 J31C ( x ) J31⁄2 = C ( x ) JJ1JJ3 + Σ | J | < | J 2 | + | J 2 ] with suitable coefficients CJ , J ... identity ( 1 ) that Σ J1 = P1 , J2 = p = Σ αγ ( α ) . | J | = 2p From this identity between formal differential operators ...
... identity 2 J31C ( x ) J31⁄2 = C ( x ) JJ1JJ3 + Σ | J | < | J 2 | + | J 2 ] with suitable coefficients CJ , J ... identity ( 1 ) that Σ J1 = P1 , J2 = p = Σ αγ ( α ) . | J | = 2p From this identity between formal differential operators ...
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 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 Haar measure 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 operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF 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