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 1142
... corollary . 13 COROLLARY . ( a ) The mapping t is a bounded map of C12 into C , 1 < p < ∞ . ( b ) If H is anti - Hermitian , then the anti - Hermitian part of TH is nH . ( c ) TH leaves invariant the range of each of the projections Ex ...
... corollary . 13 COROLLARY . ( a ) The mapping t is a bounded map of C12 into C , 1 < p < ∞ . ( b ) If H is anti - Hermitian , then the anti - Hermitian part of TH is nH . ( c ) TH leaves invariant the range of each of the projections Ex ...
Page 1301
... corollary were false , it would follow that 7 has a boundary value at a which is independent of the set A , ... , 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 7 has a boundary value at a which is independent of the set A , ... , 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 ...
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero