Linear Operators: Spectral theory |
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Page 877
... Corollary 1.10 that yy * has an inverse in . Thus y has an inverse in Y. Q.E.D. 11 COROLLARY . Let x be an element of the commutative B * -algebra X and let B * ( x ) be the smallest closed B * -subalgebra containing x and the unit e in ...
... Corollary 1.10 that yy * has an inverse in . Thus y has an inverse in Y. Q.E.D. 11 COROLLARY . Let x be an element of the commutative B * -algebra X and let B * ( x ) be the smallest closed B * -subalgebra containing x and the unit e in ...
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 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 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
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero