Linear Operators, Part 2 |
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Page 1105
... immediately from ( a ) . Thus , we have only to prove the trilinear inequality ( a ) for operators in a d - dimensional Hilbert space . Arguing as in the paragraphs of the proof of Lemma 14 following formula ( 3 ) of that proof , where ...
... immediately from ( a ) . Thus , we have only to prove the trilinear inequality ( a ) for operators in a d - dimensional Hilbert space . Arguing as in the paragraphs of the proof of Lemma 14 following formula ( 3 ) of that proof , where ...
Page 1226
... immediately from part ( a ) and Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator ... immediately from Lemma 1.6 . a To prove ( c ) , we note that since TOT , T * CT * by Lemma 1 . On the other hand , if e ...
... immediately from part ( a ) and Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator ... immediately from Lemma 1.6 . a To prove ( c ) , we note that since TOT , T * CT * by Lemma 1 . On the other hand , if e ...
Page 1746
... immediately from Theorem 23 , from the remark immediately following the proof of Theorem 23 , and from Theorems XII.2.9 ( a ) and 25. Q.E.D. The following very interesting theorem gives a general com- pleteness principle for the ...
... immediately from Theorem 23 , from the remark immediately following the proof of Theorem 23 , and from Theorems XII.2.9 ( a ) and 25. Q.E.D. The following very interesting theorem gives a general com- pleteness principle for the ...
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