Linear Operators, Part 2 |
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Page 1247
... Theorem 2.3 we have n T1 = TE ( [ — n , n ] ) = [ " ̧ λE ( d2 ) = E ( [ − n , n ] ) TE ( [ — n , n ] ) . Then ... follows from Theorem X.4.2 that T≥ 0 . Hence ( Tx , x ) = lim , ( T1x , x ) ≥ 0 and T is positive . Conversely , if T≥0 ...
... Theorem 2.3 we have n T1 = TE ( [ — n , n ] ) = [ " ̧ λE ( d2 ) = E ( [ − n , n ] ) TE ( [ — n , n ] ) . Then ... follows from Theorem X.4.2 that T≥ 0 . Hence ( Tx , x ) = lim , ( T1x , x ) ≥ 0 and T is positive . Conversely , if T≥0 ...
Page 1379
... Theorem 23 , the values ( e ) are uniquely determined for each e C N. Since A is the union of a sequence of neighborhoods of the same type as N , the uniqueness of { i } follows immediately . Q.E.D. 27 THEOREM . Let τ , T , A , 01 ...
... Theorem 23 , the values ( e ) are uniquely determined for each e C N. Since A is the union of a sequence of neighborhoods of the same type as N , the uniqueness of { i } follows immediately . Q.E.D. 27 THEOREM . Let τ , T , A , 01 ...
Page 1711
... PROOF . It is clear from Theorem XII.2.6 and from formula ( 1 ) of the proof of Theorem 6 that if f is in L2 ( I ) , then F ( T ) ƒ is in ( T ) ( T1 ( t " ) ) . Thus it follows from Corollary 5 that F ( T ) is in C ( I ) . The map f ...
... PROOF . It is clear from Theorem XII.2.6 and from formula ( 1 ) of the proof of Theorem 6 that if f is in L2 ( I ) , then F ( T ) ƒ is in ( T ) ( T1 ( t " ) ) . Thus it follows from Corollary 5 that F ( T ) is in C ( I ) . The map f ...
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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