Linear Operators: Spectral theory |
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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 Ĩ respectively . From Corollary 2.7 it is seen that VEV - 1 and hence that E = F ( T ) = VF ( T ) V - 1 for every bounded Borel function F. The mapping W = n = 1 ÜV of S onto La ( en , μ ) is clearly an isometry and ...
... identity for T and Ĩ respectively . From Corollary 2.7 it is seen that VEV - 1 and hence that E = F ( T ) = VF ( T ) V - 1 for every bounded Borel function F. The mapping W = n = 1 ÜV of S onto La ( en , μ ) is clearly an isometry and ...
Page 1717
... identity JJ 1 C ( x ) JJ z = C ( x ) JJ1JJ2 + Σ J│ < J2 + J2 with suitable coefficients CJ , J ,, holds for every ... identity ( 1 ) that Σ d12 ( x ) 31⁄231⁄2 Σ αγα ) . = | J1 = P11 | J2 = p | J | = 2p From this identity between formal ...
... identity JJ 1 C ( x ) JJ z = C ( x ) JJ1JJ2 + Σ J│ < J2 + J2 with suitable coefficients CJ , J ,, holds for every ... identity ( 1 ) that Σ d12 ( x ) 31⁄231⁄2 Σ αγα ) . = | J1 = P11 | J2 = p | J | = 2p From this identity between formal ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero